Direct sum of two arrays

Description

This function computes the direct sum of two arrays. The arrays can be numerical vectors or matrices. The result ia the block diagonal matrix.

Usage

x%s%y

Arguments

Details

If either \bf{x} or y is a vector, it is converted to a matrix. The result is a block diagonal matrix \left\lbrack {\begin{array}{cc} {\bf{x}} & {\bf{0}} \\ {\bf{0}} & {\bf{y}} \\ \end{array}} \right\rbrack.

Value

A numeric matrix.

Author(s)

Frederick Novomestky fnovomes@poly.edu, Kurt Hornik Kurt.Hornik@wu-wien.ac.at

References

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

Examples

x <- matrix( seq( 1, 4 ) )
y <- matrix( seq( 5, 8 ) )
print( x %s% y )

Duplication matrix

Description

This function constructs the linear transformation D that maps vech(A) to vec(A) when A is a symmetric matrix

Usage

D.matrix(n)

Arguments

Details

Let {\bf{T}}_{i,j} be an n \times n matrix with 1 in its \left( {i,j} \right) element 1 \le i,j \le n. and zeroes elsewhere. These matrices are constructed by the function T.matrices. The formula for the transpose of matrix \bf{D} is {\bf{D'}} = \sum\limits_{j = 1}^n {\sum\limits_{i = j}^n {{{\bf{u}}_{i,j}}\;{{\left( {vec\;{{\bf{T}}_{i,j}}} \right)}^\prime }} } where {{{\bf{u}}_{i,j}}} is the column vector in the order \frac{1}{2}n\left( {n + 1} \right) identity matrix for column k = \left( {j - 1} \right)n + i - \frac{1}{2}j\left( {j - 1} \right). The function u.vectors generates these vectors.

Value

It returns an {n^2}\; \times \;\frac{1}{2}n\left( {n + 1} \right) matrix.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1980). The elimination matrix, some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1(4), December 1980, 422-449.

Magnus, J. R. and H. Neudecker (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

See Also

T.matrices, u.vectors

Examples

D <- D.matrix( 3 )
A <- matrix( c( 1, 2, 3,
                2, 3, 4,
                3, 4, 5), nrow=3, byrow=TRUE )
vecA <- vec( A )
vechA<- vech( A )
y <- D %*% vechA
print( y )
print( vecA )

List of E Matrices

Description

This function constructs and returns a list of lists. The component of each sublist is a square matrix derived from the column vectors of an order n identity matrix.

Usage

E.matrices(n)

Arguments

Details

Let {{\bf{I}}_n} = \lbrack {\begin{array}{cccc} {{{\bf{e}}_1}}&{{{\bf{e}}_2}}& \cdots &{{{\bf{e}}_n}} \end{array}} \rbrack be the order n identity matrix with corresponding unit vectors {{{\bf{e}}_i}} with one in its ith position and zeros elsewhere. The n \times n matrix {{\bf{E}}_{i,j}} is computed from the unit vectors {{{\bf{e}}_i}} and {{{\bf{e}}_j}} as {{\bf{E}}_{i,j}} = {{\bf{e}}_i}\;{{\bf{e'}}_j}. These matrices are stored as components in a list of lists.

Value

A list with n components

...

Each component j of sublist i is a matrix {\bf{E}}_{i,j}

Note

The argument n must be an integer value greater than or equal to 2.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1980). The elimination matrix, some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1(4), December 1980, 422-449.

Magnus, J. R. and H. Neudecker (1999). Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

Examples

E <- E.matrices( 3 )

List of H Matrices

Description

This function constructs and returns a list of lists. The component of each sublist is derived from column vectors in an order r and order c identity matrix.

Usage

H.matrices(r, c = r)

Arguments

Details

Let {{\bf{I}}_r} = \lbrack {\begin{array}{cccc} {{{\bf{a}}_1}}&{{{\bf{a}}_2}}& \cdots &{{{\bf{a}}_r}} \end{array}} \rbrack be the order r identity matrix with corresponding unit vectors {{{\bf{a}}_i}} with one in its ith position and zeros elsewhere. Let {{\bf{I}}_c} = \lbrack {\begin{array}{cccc} {{{\bf{b}}_1}}&{{{\bf{b}}_2}}& \cdots &{{{\bf{b}}_c}} \end{array}} \rbrack be the order c identity matrix with corresponding unit vectors {{{\bf{b}}_i}} with one in its ith position and zeros elsewhere. The r \times c matrix {\bf{H}}{}_{i,j} = {{\bf{a}}_i}\;{{\bf{b'}}_j} is used in the computation of the commutation matrix.

Value

A list with r components

...

Each component j of sublist i is a matrix {\bf{H}}_{i,j}

Note

The argument n must be an integer value greater than or equal to two.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1979). The commutation matrix: some properties and applications, The Annals of Statistics, 7(2), 381-394.

Magnus, J. R. and H. Neudecker (1980). The elimination matrix, some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1(4), December 1980, 422-449.

Examples

H.2.3 <- H.matrices( 2, 3 )
H.3 <- H.matrices( 3 )

K Matrix

Description

This function returns a square matrix of order p = r * c that, for an r by c matrix A, transforms vec(A) to vec(A') where prime denotes transpose.

Usage

K.matrix(r, c = r)

Arguments

Details

The r \times c matrices {\bf{H}}{}_{i,j} constructed by the function H.matrices are combined using direct product to generate the commutation product with the formula {{\bf{K}}_{r,c}} = \sum\limits_{i = 1}^r {\sum\limits_{j = 1}^c {\left( {{{\bf{H}}_{i,j}} \otimes {{{\bf{H'}}}_{i,j}}} \right)} }

Value

An order \left( {r\;c} \right) matrix.

Note

If either argument is less than 2, then the function stops and displays an appropriate error mesage. If either argument is not an integer, then the function stops and displays an appropriate error mesage

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1979). The commutation matrix: some properties and applications, The Annals of Statistics, 7(2), 381-394.

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

See Also

H.matrices

Examples

K <- K.matrix( 3, 4 )
A <- matrix( seq( 1, 12, 1 ), nrow=3, byrow=TRUE )
vecA <- vec( A )
vecAt <- vec( t( A ) )
y <- K %*% vecA
print( y )
print( vecAt )

Construct L Matrix

Description

This function returns a matrix with n * ( n + 1 ) / 2 rows and N * n columns which for any lower triangular matrix A transforms vec( A ) into vech(A)

Usage

L.matrix(n)

Arguments

Details

The formula used to compute the L matrix which is also called the elimination matrix is {\bf{L}} = \sum\limits_{j = 1}^n {\sum\limits_{i = j}^n {{{\bf{u}}_{i,j}}{{\left( {vec\;{{\bf{E}}_{i,j}}} \right)}^\prime }} } {{{\bf{u}}_{i,j}}} are the n \times 1 vectors constructed by the function u.vectors. {{{\bf{E}}_{i,j}}} are the n \times n matrices constructed by the function E.matrices.

Value

An \left[ {\frac{1}{2}n\left( {n + 1} \right)} \right] \times {n^2} matrix.

Note

If the argument is not an integer, the function displays an error message and stops. If the argument is less than two, the function displays an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1980). The elimination matrix, some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1(4), December 1980, 422-449.

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

See Also

elimination.matrix, E.matrices, u.vectors,

Examples

L <- L.matrix( 4 )
A <- lower.triangle( matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE ) )
vecA <- vec( A )
vechA <- vech( A )
y <- L %*% vecA
print( y )
print( vechA )

Construct N Matrix

Description

This function returns the order n square matrix that is the sum of an implicit commutation matrix and the order n identity matrix quantity divided by two

Usage

N.matrix(n)

Arguments

Details

Let {\bf{K}_n} be the order n implicit commutation matrix (i.e., {{\bf{K}}_{n,n}} ). and {{\bf{I}}_n} the order n identity matrix. The formula for the matrix is {\bf{N}} = \frac{1}{2}\left( {{{\bf{K}}_n} + {{\bf{I}}_n}} \right).

Value

An order n matrix.

Note

If the argument is not an integer, the function displays an error message and stops. If the argument is less than two, the function displays an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1980). The elimination matrix, some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1(4), December 1980, 422-449.

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

See Also

K.matrix

Examples

N <- N.matrix( 3 )
print( N )

List of T Matrices

Description

This function constructs a list of lists. The number of components in the high level list is n. Each of the n components is also a list. Each sub-list has n components each of which is an order n square matrix.

Usage

T.matrices(n)

Arguments

Details

Let {{\bf{E}}_{i,j}}\;i = 1, \ldots ,n\;;\;j = 1, \ldots ,n be a representative order n matrix created with function E.matrices. The order n matrix {{\bf{T}}_{i,j}} is defined as follows {{\bf{T}}_{i,j}} = \left\{ {\begin{array}{cc} {{{\bf{E}}_{i,j}}}&{i = j}\\ {{{\bf{E}}_{i,j}} + {{\bf{E}}_{j,i}}}&{i \ne j} \end{array}} \right.

Value

A list of n components.

...

Each component j of sublist i is a matrix {\bf{T}}_{i,j}

Note

The argument n must be an integer value greater than or equal to 2.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1980). The elimination matrix, some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1(4), December 1980, 422-449.

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

See Also

E.matrices

Examples

T <- T.matrices( 3 )

Commutation matrix for r by c numeric matrices

Description

This function returns a square matrix of order p = r * c that, for an r by c matrix A, transforms vec(A) to vec(A') where prime denotes transpose.

Usage

commutation.matrix(r, c=r)

Arguments

Details

This function is a wrapper function that uses the function K.matrix to do the actual work. The r \times c matrices {\bf{H}}{}_{i,j} constructed by the function H.matrices are combined using direct product to generate the commutation product with the following formula {{\bf{K}}_{r,c}} = \sum\limits_{i = 1}^r {\sum\limits_{j = 1}^c {\left( {{{\bf{H}}_{i,j}} \otimes {{{\bf{H'}}}_{i,j}}} \right)} }

Value

An order \left( {r\;c} \right) matrix.

Note

If either argument is less than 2, then the function stops and displays an appropriate error mesage. If either argument is not an integer, then the function stops and displays an appropriate error mesage

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1979). The commutation matrix: some properties and applications, The Annals of Statistics, 7(2), 381-394.

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

See Also

H.matrices, K.matrix

Examples

K <- commutation.matrix( 3, 4 )
A <- matrix( seq( 1, 12, 1 ), nrow=3, byrow=TRUE )
vecA <- vec( A )
vecAt <- vec( t( A ) )
print( K %*% vecA )
print( vecAt )

Creation Matrix

Description

This function returns the order n creation matrix, a square matrix with the sequence 1, 2, ..., n - 1 on the sub-diagonal below the principal diagonal.

Usage

creation.matrix(n)

Arguments

Details

The order n creation matrix is also called the derivation matrix and is used in numerical mathematics and physics. It arises in the solution of linear dynamical systems. The form of the matrix is \left\lbrack {\begin{array}{cccccc} 0&0&0& \cdots &0&0\\ 1&0&0& \cdots &0&0\\ 0&2&0& \cdots &0&0\\ 0&0&3& \ddots &0&0\\ \vdots & \vdots & \vdots & \ddots & \ddots &{}\\ 0&0&0& \cdots &{n - 1}&0 \end{array}} \right\rbrack.

Value

An order n matrix.

Note

If the argument n is not an integer that is greater than 1, the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Aceto, L. and D. Trigiante (2001). Matrices of Pascal and Other Greats, American Mathematical Monthly, March 2001, 108(3), 232-245.

Weinberg, S. (1995). The Quantum Theory of Fields, Cambridge University Press.

Examples

H <- creation.matrix( 10 )
print( H )

Direct prod of two arrays

Description

This function computes the direct product of two arrays. The arrays can be numerical vectors or matrices. The result is a matrix.

Usage

direct.prod( x, y )

Arguments

Details

If either \bf{x} or \bf{y} is a vector, it is converted to a matrix. Suppose that \bf{x} is an m \times n matrix and \bf{y} is an p \times q matrix. Then, the function returns the matrix \left\lbrack {\begin{array}{cccc} {{x_{1,1}}\;{\bf{y}}}&{{x_{1,2}}\;{\bf{y}}}& \cdots &{{x_{1,n}}\;{\bf{y}}}\\ {{x_{2,1}}\;{\bf{y}}}&{{x_{2,2}}\;{\bf{y}}}& \cdots &{{x_{2,n}}\;{\bf{y}}}\\ \cdots & \cdots & \cdots & \cdots \\ {{x_{m,1}}\;{\bf{y}}}&{{x_{m,2}}\;{\bf{y}}}& \cdots &{{x_{m,n}}\;{\bf{y}}} \end{array}} \right\rbrack.

Value

A numeric matrix.

Author(s)

Frederick Novomestky fnovomes@poly.edu, Kurt Hornik Kurt.Hornik@wu-wien.ac.at

References

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

Examples

x <- matrix( seq( 1, 4 ) )
y <- matrix( seq( 5, 8 ) )
print( direct.prod( x, y ) )

Direct sum of two arrays

Description

This function computes the direct sum of two arrays. The arrays can be numerical vectors or matrices. The result ia the block diagonal matrix.

Usage

direct.sum( x, y )

Arguments

Details

If either \bf{x} or y is a vector, it is converted to a matrix. The result is a block diagonal matrix \left\lbrack {\begin{array}{cc} {\bf{x}} & {\bf{0}} \\ {\bf{0}} & {\bf{y}} \\ \end{array}} \right\rbrack.

Value

A numeric matrix.

Author(s)

Frederick Novomestky fnovomes@poly.edu, Kurt Hornik Kurt.Hornik@wu-wien.ac.at

References

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

Examples

x <- matrix( seq( 1, 4 ) )
y <- matrix( seq( 5, 8 ) )
print( direct.sum( x, y ) )

Duplication matrix for n by n matrices

Description

This function returns a matrix with n * n rows and n * ( n + 1 ) / 2 columns that transforms vech(A) to vec(A) where A is a symmetric n by n matrix.

Usage

duplication.matrix(n=1)

Arguments

Details

This function is a wrapper function for the function D.matrix. Let {\bf{T}}_{i,j} be an n \times n matrix with 1 in its \left( {i,j} \right) element 1 \le i,j \le n. and zeroes elsewhere. These matrices are constructed by the function T.matrices. The formula for the transpose of matrix \bf{D} is {\bf{D'}} = \sum\limits_{j = 1}^n {\sum\limits_{i = j}^n {{{\bf{u}}_{i,j}}\;{{\left( {vec\;{{\bf{T}}_{i,j}}} \right)}^\prime }} } where {{{\bf{u}}_{i,j}}} is the column vector in the order \frac{1}{2}n\left( {n + 1} \right) identity matrix for column k = \left( {j - 1} \right)n + i - \frac{1}{2}j\left( {j - 1} \right). The function u.vectors generates these vectors.

Value

It returns an {n^2}\; \times \;\frac{1}{2}n\left( {n + 1} \right) matrix.

Author(s)

Frederick Novomestky fnovomes@poly.edu, Kurt Hornik Kurt.Hornik@wu-wien.ac.at

References

Magnus, J. R. and H. Neudecker (1980). The elimination matrix, some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1(4), December 1980, 422-449.

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

See Also

D.matrix, vec, vech

Examples

D <- duplication.matrix( 3 )
A <- matrix( c( 1, 2, 3,
                2, 3, 4,
                3, 4, 5), nrow=3, byrow=TRUE )
vecA <- vec( A )
vechA<- vech( A )
y <- D %*% vechA
print( y )
print( vecA )

Elimination matrix for lower triangular matrices

Description

This function returns a matrix with n * ( n + 1 ) / 2 rows and N * n columns which for any lower triangular matrix A transforms vec( A ) into vech(A)

Usage

elimination.matrix(n)

Arguments

Details

This function is a wrapper function to the function L.matrix. The formula used to compute the L matrix which is also called the elimination matrix is {\bf{L}} = \sum\limits_{j = 1}^n {\sum\limits_{i = j}^n {{{\bf{u}}_{i,j}}{{\left( {vec\;{{\bf{E}}_{i,j}}} \right)}^\prime }} } {{{\bf{u}}_{i,j}}} are the order n\left( {n + 1} \right)/2 vectors constructed by the function u.vectors. {{{\bf{E}}_{i,j}}} are the n \times n matrices constructed by the function E.matrices.

Value

An \left[ {\frac{1}{2}n\left( {n + 1} \right)} \right] \times {n^2} matrix.

Note

If the argument is not an integer, the function displays an error message and stops. If the argument is less than two, the function displays an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1980). The elimination matrix, some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1(4), December 1980, 422-449.

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

See Also

E.matrices, L.matrix, u.vectors

Examples

L <- elimination.matrix( 4 )
A <- lower.triangle( matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE ) )
vecA <- vec( A )
vechA <- vech( A )
y <- L %*% vecA
print( y )
print( vechA )

Compute the entrywise norm of a matrix

Description

This function returns the \left\| {\bf{x}} \right\|_p norm of the matrix {\mathbf{x}}.

Usage

entrywise.norm(x,p)

Arguments

Details

Let {\bf{x}} be an m \times n numeric matrix. The formula used to compute the norm is \left\| {\bf{x}} \right\|_p = \left( {\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {\left| {x_{i,j} } \right|^p } } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. } p}}.

Value

A numeric value.

Note

If argument x is not numeric, the function displays an error message and terminates. If argument x is neither a matrix nor a vector, the function displays an error message and terminates. If argument p is zero, the function displays an error message and terminates.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Golub, G. H. and C. F. Van Loan (1996). Matrix Computations, Third Edition, The John Hopkins University Press.

Horn, R. A. and C. R. Johnson (1985). Matrix Analysis, Cambridge University Press.

See Also

one.norm, inf.norm

Examples

A <- matrix( c( 3, 5, 7, 2, 6, 4, 0, 2, 8 ), nrow=3, ncol=3, byrow=TRUE )
print( entrywise.norm( A, 2 ) )

Fibonacci Matrix

Description

This function constructs the order n + 1 square Fibonacci matrix which is derived from a Fibonacci sequence.

Usage

fibonacci.matrix(n)

Arguments

Details

Let \left\{ {{f_0},\;{f_1},\; \ldots ,\;{f_n}} \right\} be the set of n + 1 Fibonacci numbers where {f_0} = {f_1} = 1 and {f_j} = {f_{j - 1}} + {f_{j - 2}},\quad 2 \le j \le n. The order n + 1 Fibonacci matrix {\bf{F}} has as typical element {F_{i,j}} = \left\{ {\begin{array}{cc} {{f_{i - j + 1}}}&{i - j + 1 \ge 0}\\ 0&{i - j + 1 < 0} \end{array}} \right..

Value

An order n + 1 matrix

Note

If the argument n is not a positive integer, the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Zhang, Z. and J. Wang (2006). Bernoulli matrix and its algebraic properties, Discrete Applied Nathematics, 154, 1622-1632.

Examples

F <- fibonacci.matrix( 10 )
print( F )

Frobenius Matrix

Description

This function returns an order n Frobenius matrix that is useful in numerical mathematics.

Usage

frobenius.matrix(n)

Arguments

Details

The Frobenius matrix is also called the companion matrix. It arises in the solution of systems of linear first order differential equations. The formula for the order n Frobenius matrix is {\bf{F}} = \left\lbrack {\begin{array}{ccccc}0&0& \cdots &0&{{{\left( { - 1} \right)}^{n - 1}} \left( {\begin{array}{ccccc}n\\0\end{array}} \right)}\\1&0& \cdots &0&{{{\left( { - 1} \right)}^{n - 2}} \left( {\begin{array}{ccccc}n\\1\end{array}} \right)}\\0&1& \ddots &0&{{{\left( { - 1} \right)}^{n - 3}} \left( {\begin{array}{ccccc}n\\2\end{array}} \right)}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\0&0& \cdots &1&{{{\left( { - 1} \right)}^0} \left( {\begin{array}{ccccc}n\\{n - 1}\end{array}} \right)}\end{array}} \right\rbrack.

Value

An order n matrix

Note

If the argument n is not a positive integer that is greater than 1, the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Aceto, L. and D. Trigiante (2001). Matrices of Pascal and Other Greats, American Mathematical Monthly, March 2001, 108(3), 232-245.

Examples

F <- frobenius.matrix( 10 )
print( F )

Compute the Frobenius norm of a matrix

Description

This function returns the Frobenius norm of the matrix {\mathbf{x}}.

Usage

frobenius.norm(x)

Arguments

Details

The formula used to compute the norm is \left\| {\bf{x}} \right\|_2. Note that this is the entrywise norm with exponent 2.

Value

A numeric value.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Golub, G. H. and C. F. Van Loan (1996). Matrix Computations, Third Edition, The John Hopkins University Press.

Horn, R. A. and C. R. Johnson (1985). Matrix Analysis, Cambridge University Press.

See Also

entrywise.norm

Examples

A <- matrix( c( 3, 5, 7, 2, 6, 4, 0, 2, 8 ), nrow=3, ncol=3, byrow=TRUE )
print( frobenius.norm( A ) )

Frobenius innter product of matrices

Description

This function returns the Fronbenius inner product of two matrices, x and y, with the same row and column dimensions.

Usage

frobenius.prod(x, y)

Arguments

Details

The Frobenius inner product is the element-by-element sum of the Hadamard or Shur product of two numeric matrices. Let {\bf{x}} and {\bf{y}} be two m \times n matrices. Then Frobenious inner product is computed as \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {x_{i,j} \;y_{i,j} } } .

Value

A numeric value.

Note

The function converts vectors to matrices if necessary. The function stops running if x or y is not numeric and an error message is displayed. The function also stops running if x and y do not have the same row and column dimensions and an error mesage is displayed.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Styan, G. P. H. (1973). Hadamard Products and Multivariate Statistical Analysis, Linear Algebra and Its Applications, Elsevier, 6, 217-240.

See Also

hadamard.prod

Examples

x <- matrix( c( 1, 2, 3, 4 ), nrow=2, byrow=TRUE )
y <- matrix( c( 2, 4, 6, 8 ), nrow=2, byrow=TRUE )
z <- frobenius.prod( x, y )
print( z )

Hadamard product of two matrices

Description

This function returns the Hadamard or Shur product of two matrices, x and y, that have the same row and column dimensions.

Usage

hadamard.prod(x, y)

Arguments

Details

The Hadamard product is an element-by-element product of the two matrices. Let {\bf{x}} and {\bf{x}} be two m \times n numeric matrices. The Hadamard product is {\bf{x}}\, \circ \,{\bf{y}} = \left\lbrack {\begin{array}{cccc} {{x_{1,1}}\,{y_{1,1}}}&{{x_{1,2}}\,{y_{1,2}}}& \cdots &{{x_{1,n}}\,{y_{1,n}}}\\ {{x_{2,1}}\,{y_{121}}}&{{x_{2,2}}\,{y_{2,2}}}& \cdots &{{x_{2,n}}\,{y_{2,n}}}\\ \cdots & \cdots & \cdots & \cdots \\ {{x_{m,1}}\,{y_{m,1}}}&{{x_{m,2}}\,{y_{m,2}}}& \cdots &{{x_{m,n}}\,{y_{m,n}}} \end{array}} \right\rbrack. It uses the * operation in R.

Value

A matrix.

Note

The function converts vectors to matrices if necessary. The function stops running if x or y is not numeric and an error message is displayed. The function also stops running if x and y do not have the same row and column dimensions and an error mesage is displayed.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Hadamard, J (1983). Resolution d'une question relative aux determinants, Bulletin des Sciences Mathematiques, 17, 240-246.

Styan, G. P. H. (1973). Hadamard Products and Multivariate Statistical Analysis, Linear Algebra and Its Applications, Elsevier, 6, 217-240.

Examples

x <- matrix( c( 1, 2, 3, 4 ), nrow=2, byrow=TRUE )
y <- matrix( c( 2, 4, 6, 8 ), nrow=2, byrow=TRUE )
z <- hadamard.prod( x, y )
print( z )

Hankel Matrix

Description

This function constructs an order n Hankel matrix from the values in the order n vector x. Each row of the matrix is a circular shift of the values in the previous row.

Usage

hankel.matrix(n, x)

Arguments

Details

A Hankel matrix is a square matrix with constant skew diagonals. The determinant of a Hankel matrix is called a catalecticant. Hankel matrices are formed when the hidden Mark model is sought from a given sequence of data.

Value

An order n matrix.

Note

If the argument n is not a positive integer, the function presents an error message and stops. If the length of x is less than n, the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Power, S. C. (1982). Hankel Operators on Hilbert Spaces, Research notes in mathematics, Series 64, Pitman Publishing.

Examples

H <- hankel.matrix( 4, seq( 1, 7 ) )
print( H )

Hilbert matrices

Description

This function returns an n by n Hilbert matrix.

Usage

hilbert.matrix(n)

Arguments

Details

A Hilbert matrix is an order n square matrix of unit fractions with elements defined as H_{i,j} = {1 \mathord{\left/ {\vphantom {1 {\left( {i + j - 1} \right)}}} \right. } {\left( {i + j - 1} \right)}}.

Value

A matrix.

Note

If the argument is less than or equal to zero, the function displays an error message and stops. If the argument is not an integer, the function displays an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Hilbert, David (1894). Ein Beitrag zur Theorie des Legendre schen Polynoms, Acta Mathematica, Springer, Netherlands, 18, 155-159.

Examples

H <- hilbert.matrix( 4 )
print( H )

Compute the Hilbert-Schmidt norm of a matrix

Description

This function returns the Hilbert-Schmidt norm of the matrix {\mathbf{x}}.

Usage

hilbert.schmidt.norm(x)

Arguments

Details

The formula used to compute the norm is \left\| {\bf{x}} \right\|_2. This is merely the entrywise norm with exponent 2.

Value

A numeric value.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Golub, G. H. and C. F. Van Loan (1996). Matrix Computations, Third Edition, The John Hopkins University Press.

Horn, R. A. and C. R. Johnson (1985). Matrix Analysis, Cambridge University Press.

See Also

entrywise.norm

Examples

A <- matrix( c( 3, 5, 7, 2, 6, 4, 0, 2, 8 ), nrow=3, ncol=3, byrow=TRUE )
print( hilbert.schmidt.norm( A ) )

Compute the infinitity norm of a matrix

Description

This function returns the \left\| {\mathbf{x}} \right\|_\infty norm of the matrix {\mathbf{x}}.

Usage

inf.norm(x)

Arguments

Details

Let {\bf{x}} be an m \times n numeric matrix. The formula used to compute the norm is \left\| {\bf{x}} \right\|_\infty = \mathop {\max }\limits_{1 \le i \le m} \sum\limits_{j = 1}^n {\left| {x_{i,j} } \right|} . This is merely the maximum absolute row sum of the m \times n maxtris.

Value

A numeric value.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Golub, G. H. and C. F. Van Loan (1996). Matrix Computations, Third Edition, The John Hopkins University Press.

Horn, R. A. and C. R. Johnson (1985). Matrix Analysis, Cambridge University Press.

See Also

one.norm

Examples

A <- matrix( c( 3, 5, 7, 2, 6, 4, 0, 2, 8 ), nrow=3, ncol=3, byrow=TRUE )
print( inf.norm( A ) )

Test for diagonal square matrix

Description

This function returns TRUE if the given matrix argument x is a square numeric matrix and that the off-diagonal elements are close to zero in absolute value to within the given tolerance level. Otherwise, a FALSE value is returned.

Usage

is.diagonal.matrix(x, tol = 1e-08)

Arguments

Value

A TRUE or FALSE value.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Horn, R. A. and C. R. Johnson (1990). Matrix Analysis, Cambridge University Press.

Examples

A <- diag( 1, 3 )
is.diagonal.matrix( A )
B <- matrix( c( 1, 2, 3, 4 ), nrow=2, byrow=TRUE )
is.diagonal.matrix( B )
C <- matrix( c( 1, 0, 0, 0 ), nrow=2, byrow=TRUE )
is.diagonal.matrix( C )

Test for idempotent square matrix

Description

This function returns a TRUE value if the square matrix argument x is idempotent, that is, the product of the matrix with itself is the matrix. The equality test is performed to within the specified tolerance level. If the matrix is not idempotent, then a FALSE value is returned.

Usage

is.idempotent.matrix(x, tol = 1e-08)

Arguments

Details

Idempotent matrices are used in econometric analysis. Consider the problem of estimating the regression parameters of a standard linear model {\bf{y}} = {\bf{X}}\;{\bf{\beta }} + {\bf{e}} using the method of least squares. {\bf{y}} is an order m random vector of dependent variables. {\bf{X}} is an m \times n matrix whose columns are columns of observations on one of the n - 1 independent variables. The first column contains m ones. {\bf{e}} is an order m random vector of zero mean residual values. {\bf{\beta }} is the order n vector of regression parameters. The objective function that is minimized in the method of least squares is \left( {{\bf{y}} - {\bf{X}}\;{\bf{\beta }}} \right)^\prime \left( {{\bf{y}} - {\bf{X}}\;{\bf{\beta }}} \right). The solution to ths quadratic programming problem is {\bf{\hat \beta }} = \left[ {\left( {{\bf{X'}}\;{\bf{X}}} \right)^{ - 1} \;{\bf{X'}}} \right]\;{\bf{y}} The corresponding estimator for the residual vector is {\bf{\hat e}} = {\bf{y}} - {\bf{X}}\;{\bf{\hat \beta }} = \left[ {{\bf{I}} - {\bf{X}}\;\left( {{\bf{X'}}\;{\bf{X}}} \right)^{ - 1} {\bf{X'}}} \right]{\bf{y}} = {\bf{M}}\;{\bf{y}}. {\bf{M}} and {{\bf{X}}\;\left( {{\bf{X'}}\;{\bf{X}}} \right)^{ - 1} {\bf{X'}}} are idempotent. Idempotency of {\bf{M}} enters into the estimation of the variance of the estimator.

Value

A TRUE or FALSE value.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Chang, A. C., (1984). Fundamental Methods of Mathematical Economics, Third edition, McGraw-Hill.

Green, W. H. (2003). Econometric Analysis, Fifth edition, Prentice-Hall.

Horn, R. A. and C. R. Johnson (1990). Matrix Analysis, Cambridge University Press.

Examples

A <- diag( 1, 3 )
is.idempotent.matrix( A )
B <- matrix( c( 1, 2, 3, 4 ), nrow=2, byrow=TRUE )
is.idempotent.matrix( B )
C <- matrix( c( 1, 0, 0, 0 ), nrow=2, byrow=TRUE )
is.idempotent.matrix( C )

Test matrix for positive indefiniteness

Description

This function returns TRUE if the argument, a square symmetric real matrix x, is indefinite. That is, the matrix has both positive and negative eigenvalues.

Usage

is.indefinite(x, tol=1e-8)

Arguments

Details

For an indefinite matrix, the matrix should positive and negative eigenvalues. The R function eigen is used to compute the eigenvalues. If any of the eigenvalues is absolute value is less than the given tolerance, that eigenvalue is replaced with zero. If the matrix has both positive and negative eigenvalues, it is declared to be indefinite.

Value

TRUE or FALSE.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

See Also

is.positive.definite, is.positive.semi.definite, is.negative.definite, is.negative.semi.definite

Examples

###
### identity matrix is always positive definite
###
I <- diag( 1, 3 )
is.indefinite( I )
###
### positive definite matrix
### eigenvalues are 3.4142136 2.0000000 0.585786
###
A <- matrix( c( 2, -1, 0, -1, 2, -1, 0, -1, 2 ), nrow=3, byrow=TRUE )
is.indefinite( A )
###
### positive semi-defnite matrix
### eigenvalues are 4.732051 1.267949 8.881784e-16
###
B <- matrix( c( 2, -1, 2, -1, 2, -1, 2, -1, 2 ), nrow=3, byrow=TRUE )
is.indefinite( B )
###
### negative definite matrix
### eigenvalues are -0.5857864 -2.0000000 -3.4142136
###
C <- matrix( c( -2, 1, 0, 1, -2, 1, 0, 1, -2 ), nrow=3, byrow=TRUE )
is.indefinite( C )
###
### negative semi-definite matrix
### eigenvalues are 1.894210e-16 -1.267949 -4.732051
###
D <- matrix( c( -2, 1, -2, 1, -2, 1, -2, 1, -2 ), nrow=3, byrow=TRUE )
is.indefinite( D )
###
### indefinite matrix
### eigenvalues are 3.828427  1.000000 -1.828427
###
E <- matrix( c( 1, 2, 0, 2, 1, 2, 0, 2, 1 ), nrow=3, byrow=TRUE )
is.indefinite( E )

Test matrix for negative definiteness

Description

This function returns TRUE if the argument, a square symmetric real matrix x, is negative definite.

Usage

is.negative.definite(x, tol=1e-8)

Arguments

Details

For a negative definite matrix, the eigenvalues should be negative. The R function eigen is used to compute the eigenvalues. If any of the eigenvalues in absolute value is less than the given tolerance, that eigenvalue is replaced with zero. If any of the eigenvalues is greater than or equal to zero, then the matrix is not negative definite. Otherwise, the matrix is declared to be negative definite.

Value

TRUE or FALSE.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

See Also

is.positive.definite, is.positive.semi.definite, is.negative.semi.definite, is.indefinite

Examples

###
### identity matrix is always positive definite
I <- diag( 1, 3 )
is.negative.definite( I )
###
### positive definite matrix
### eigenvalues are 3.4142136 2.0000000 0.585786
###
A <- matrix( c( 2, -1, 0, -1, 2, -1, 0, -1, 2 ), nrow=3, byrow=TRUE )
is.negative.definite( A )
###
### positive semi-defnite matrix
### eigenvalues are 4.732051 1.267949 8.881784e-16
###
B <- matrix( c( 2, -1, 2, -1, 2, -1, 2, -1, 2 ), nrow=3, byrow=TRUE )
is.negative.definite( B )
###
### negative definite matrix
### eigenvalues are -0.5857864 -2.0000000 -3.4142136
###
C <- matrix( c( -2, 1, 0, 1, -2, 1, 0, 1, -2 ), nrow=3, byrow=TRUE )
is.negative.definite( C )
###
### negative semi-definite matrix
### eigenvalues are 1.894210e-16 -1.267949 -4.732051
###
D <- matrix( c( -2, 1, -2, 1, -2, 1, -2, 1, -2 ), nrow=3, byrow=TRUE )
is.negative.definite( D )
###
### indefinite matrix
### eigenvalues are 3.828427  1.000000 -1.828427
###
E <- matrix( c( 1, 2, 0, 2, 1, 2, 0, 2, 1 ), nrow=3, byrow=TRUE )
is.negative.definite( E )

Test matrix for negative semi definiteness

Description

This function returns TRUE if the argument, a square symmetric real matrix x, is negative semi-negative.

Usage

is.negative.semi.definite(x, tol=1e-8)

Arguments

Details

For a negative semi-definite matrix, the eigenvalues should be non-positive. The R function eigen is used to compute the eigenvalues. If any of the eigenvalues in absolute value is less than the given tolerance, that eigenvalue is replaced with zero. Then, if any of the eigenvalues is greater than zero, the matrix is not negative semi-definite. Otherwise, the matrix is declared to be negative semi-definite.

Value

TRUE or FALSE.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

See Also

is.positive.definite, is.positive.semi.definite, is.negative.definite, is.indefinite

Examples

###
### identity matrix is always positive definite
I <- diag( 1, 3 )
is.negative.semi.definite( I )
###
### positive definite matrix
### eigenvalues are 3.4142136 2.0000000 0.585786
###
A <- matrix( c( 2, -1, 0, -1, 2, -1, 0, -1, 2 ), nrow=3, byrow=TRUE )
is.negative.semi.definite( A )
###
### positive semi-defnite matrix
### eigenvalues are 4.732051 1.267949 8.881784e-16
###
B <- matrix( c( 2, -1, 2, -1, 2, -1, 2, -1, 2 ), nrow=3, byrow=TRUE )
is.negative.semi.definite( B )
###
### negative definite matrix
### eigenvalues are -0.5857864 -2.0000000 -3.4142136
###
C <- matrix( c( -2, 1, 0, 1, -2, 1, 0, 1, -2 ), nrow=3, byrow=TRUE )
is.negative.semi.definite( C )
###
### negative semi-definite matrix
### eigenvalues are 1.894210e-16 -1.267949 -4.732051
###
D <- matrix( c( -2, 1, -2, 1, -2, 1, -2, 1, -2 ), nrow=3, byrow=TRUE )
is.negative.semi.definite( D )
###
### indefinite matrix
### eigenvalues are 3.828427  1.000000 -1.828427
###
E <- matrix( c( 1, 2, 0, 2, 1, 2, 0, 2, 1 ), nrow=3, byrow=TRUE )
is.negative.semi.definite( E )

Test if matrix is non-singular

Description

This function returns TRUE is the matrix argument is non-singular and FALSE otherwise.

Usage

is.non.singular.matrix(x, tol = 1e-08)

Arguments

Details

The determinant of the matrix x is first computed. If the absolute value of the determinant is greater than or equal to the given tolerance level, then a TRUE value is returned. Otherwise, a FALSE value is returned.

Value

TRUE or FALSE value.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Horn, R. A. and C. R. Johnson (1990). Matrix Analysis, Cambridge University Press.

See Also

is.singular.matrix

Examples

A <- diag( 1, 3 )
is.non.singular.matrix( A )
B <- matrix( c( 0, 0, 3, 4 ), nrow=2, byrow=TRUE )
is.non.singular.matrix( B )

Test matrix for positive definiteness

Description

This function returns TRUE if the argument, a square symmetric real matrix x, is positive definite.

Usage

is.positive.definite(x, tol=1e-8)

Arguments

Details

For a positive definite matrix, the eigenvalues should be positive. The R function eigen is used to compute the eigenvalues. If any of the eigenvalues in absolute value is less than the given tolerance, that eigenvalue is replaced with zero. If any of the eigenvalues is less than or equal to zero, then the matrix is not positive definite. Otherwise, the matrix is declared to be positive definite.

Value

TRUE or FALSE.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

See Also

is.positive.semi.definite, is.negative.definite, is.negative.semi.definite, is.indefinite

Examples

###
### identity matrix is always positive definite
I <- diag( 1, 3 )
is.positive.definite( I )
###
### positive definite matrix
### eigenvalues are 3.4142136 2.0000000 0.585786
###
A <- matrix( c( 2, -1, 0, -1, 2, -1, 0, -1, 2 ), nrow=3, byrow=TRUE )
is.positive.definite( A )
###
### positive semi-defnite matrix
### eigenvalues are 4.732051 1.267949 8.881784e-16
###
B <- matrix( c( 2, -1, 2, -1, 2, -1, 2, -1, 2 ), nrow=3, byrow=TRUE )
is.positive.definite( B )
###
### negative definite matrix
### eigenvalues are -0.5857864 -2.0000000 -3.4142136
###
C <- matrix( c( -2, 1, 0, 1, -2, 1, 0, 1, -2 ), nrow=3, byrow=TRUE )
is.positive.definite( C )
###
### negative semi-definite matrix
### eigenvalues are 1.894210e-16 -1.267949 -4.732051
###
D <- matrix( c( -2, 1, -2, 1, -2, 1, -2, 1, -2 ), nrow=3, byrow=TRUE )
is.positive.definite( D )
###
### indefinite matrix
### eigenvalues are 3.828427  1.000000 -1.828427
###
E <- matrix( c( 1, 2, 0, 2, 1, 2, 0, 2, 1 ), nrow=3, byrow=TRUE )
is.positive.definite( E )

Test matrix for positive semi-definiteness

Description

This function returns TRUE if the argument, a square symmetric real matrix x, is positive semi-definite.

Usage

is.positive.semi.definite(x, tol=1e-8)

Arguments

Details

For a positive semi-definite matrix, the eigenvalues should be non-negative. The R function eigen is used to compute the eigenvalues. If any of the eigenvalues is less than zero, then the matrix is not positive semi-definite. Otherwise, the matrix is declared to be positive semi-definite.

Value

TRUE or FALSE.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

See Also

is.positive.definite, is.negative.definite, is.negative.semi.definite, is.indefinite

Examples

###
### identity matrix is always positive definite
I <- diag( 1, 3 )
is.positive.semi.definite( I )
###
### positive definite matrix
### eigenvalues are 3.4142136 2.0000000 0.585786
###
A <- matrix( c( 2, -1, 0, -1, 2, -1, 0, -1, 2 ), nrow=3, byrow=TRUE )
is.positive.semi.definite( A )
###
### positive semi-defnite matrix
### eigenvalues are 4.732051 1.267949 8.881784e-16
###
B <- matrix( c( 2, -1, 2, -1, 2, -1, 2, -1, 2 ), nrow=3, byrow=TRUE )
is.positive.semi.definite( B )
###
### negative definite matrix
### eigenvalues are -0.5857864 -2.0000000 -3.4142136
###
C <- matrix( c( -2, 1, 0, 1, -2, 1, 0, 1, -2 ), nrow=3, byrow=TRUE )
is.positive.semi.definite( C )
###
### negative semi-definite matrix
### eigenvalues are 1.894210e-16 -1.267949 -4.732051
###
D <- matrix( c( -2, 1, -2, 1, -2, 1, -2, 1, -2 ), nrow=3, byrow=TRUE )
is.positive.semi.definite( D )
###
### indefinite matrix
### eigenvalues are 3.828427  1.000000 -1.828427
###
E <- matrix( c( 1, 2, 0, 2, 1, 2, 0, 2, 1 ), nrow=3, byrow=TRUE )
is.positive.semi.definite( E )

Test for singular square matrix

Description

This function returns TRUE is the matrix argument is singular and FALSE otherwise.

Usage

is.singular.matrix(x, tol = 1e-08)

Arguments

Details

The determinant of the matrix x is first computed. If the absolute value of the determinant is less than the given tolerance level, then a TRUE value is returned. Otherwise, a FALSE value is returned.

Value

A TRUE or FALSE value.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Horn, R. A. and C. R. Johnson (1990). Matrix Analysis, Cambridge University Press.

See Also

is.non.singular.matrix

Examples

A <- diag( 1, 3 )
is.singular.matrix( A )
B <- matrix( c( 0, 0, 3, 4 ), nrow=2, byrow=TRUE )
is.singular.matrix( B )

Test for a skew-symmetric matrix

Description

This function returns TRUE if the matrix argument x is a skew symmetric matrix, i.e., the transpose of the matrix is the negative of the matrix. Otherwise, FALSE is returned.

Usage

is.skew.symmetric.matrix(x, tol = 1e-08)

Arguments

Details

Let {\bf{x}} be an order n matrix. If every element of the matrix {\bf{x}} + {\bf{x'}} in absolute value is less than the given tolerance, then the matrix argument is declared to be skew symmetric.

Value

A TRUE or FALSE value.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Horn, R. A. and C. R. Johnson (1990). Matrix Analysis, Cambridge University Press.

Examples

A <- diag( 1, 3 )
is.skew.symmetric.matrix( A )
B <- matrix( c( 0, -2, -1, -2, 0, -4, 1, 4, 0 ), nrow=3, byrow=TRUE )
is.skew.symmetric.matrix( B )
C <- matrix( c( 0, 2, 1, 2, 0, 4, 1, 4, 0 ), nrow=3, byrow=TRUE )
is.skew.symmetric.matrix( C )

Test for square matrix

Description

The function returns TRUE if the argument is a square matrix and FALSE otherwise.

Usage

is.square.matrix(x)

Arguments

Value

TRUE or FALSE

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Examples

A <- matrix( seq( 1, 12, 1 ), nrow=3, byrow=TRUE )
is.square.matrix( A )
B <- matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE )
is.square.matrix( B )

Test for symmetric numeric matrix

Description

This function returns TRUE if the argument is a numeric symmetric square matrix and FALSE otherwise.

Usage

is.symmetric.matrix(x)

Arguments

Value

TRUE or FALSE.

Note

If the argument is not a numeric matrix, the function displays an error message and stops. If the argument is not a square matrix, the function displays an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

See Also

is.square.matrix

Examples

A <- matrix( c( 1, 2, 3, 4 ), nrow=2, byrow=TRUE )
is.symmetric.matrix( A )
B <- matrix( c( 1, 2, 2, 1 ), nrow=2, byrow=TRUE )
is.symmetric.matrix( B )

Lower triangle portion of a matrix

Description

Returns the lower triangle including the diagonal of a square numeric matrix.

Usage

lower.triangle(x)

Arguments

Value

A matrix.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

See Also

is.square.matrix

Examples

B <- matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE )
lower.triangle( B )

LU Decomposition of Square Matrix

Description

This function performs an LU decomposition of the given square matrix argument the results are returned in a list of named components. The Doolittle decomposition method is used to obtain the lower and upper triangular matrices

Usage

lu.decomposition(x)

Arguments

Details

The Doolittle decomposition without row exchanges is performed generating the lower and upper triangular matrices separately rather than in one matrix.

Value

A list with two named components.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Golub, G. H. and C. F. Van Loan (1996). Matrix Computations, Third Edition, John Hopkins University Press

Horn, R. A. and C. R. Johnson (1985). Matrix Analysis, Cambridge University Press.

Examples

A <- matrix( c ( 1, 2, 2, 1 ), nrow=2, byrow=TRUE)
luA <- lu.decomposition( A )
L <- luA$L
U <- luA$U
print( L )
print( U )
print( L %*% U )
print( A )
B <- matrix( c( 2, -1, -2, -4, 6, 3, -4, -2, 8 ), nrow=3, byrow=TRUE )
luB <- lu.decomposition( B )
L <- luB$L
U <- luB$U
print( L )
print( U )
print( L %*% U )
print( B )

Inverse of a square matrix

Description

This function returns the inverse of a square matrix computed using the R function solve.

Usage

matrix.inverse(x)

Arguments

Value

A matrix.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Examples

A <- matrix( c ( 1, 2, 2, 1 ), nrow=2, byrow=TRUE)
print( A )
invA <- matrix.inverse( A )
print( invA )
print( A %*% invA )
print( invA %*% A )

Matrix Raised to a Power

Description

This function computes the k-th power of order n square matrix x If k is zero, the order n identity matrix is returned. argument k must be an integer.

Usage

matrix.power(x, k)

Arguments

Details

The matrix power is computed by successive matrix multiplications. If the exponent is zero, the order n identity matrix is returned. If the exponent is negative, the inverse of the matrix is raised to the given power.

Value

An order n matrix.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Examples

A <- matrix( c ( 1, 2, 2, 1 ), nrow=2, byrow=TRUE)
matrix.power( A, -2 )
matrix.power( A, -1 )
matrix.power( A, 0 )
matrix.power( A, 1 )
matrix.power( A, 2 )

Rank of a square matrix

Description

This function returns the rank of a square numeric matrix based on the selected method.

Usage

matrix.rank(x, method = c("qr", "chol"))

Arguments

Details

If the user specifies "qr" as the method, then the QR decomposition function is used to obtain the rank. If the user specifies "chol" as the method, the rank is obtained from the attributes of the value returned.

Value

An integer.

Note

If the argument is not a square numeric matrix, then the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

See Also

is.square.matrix

Examples

A <- diag( seq( 1, 4, 1 ) )
matrix.rank( A )
B <- matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE )
matrix.rank( B )

The trace of a matrix

Description

This function returns the trace of a given square numeric matrix.

Usage

matrix.trace(x)

Arguments

Value

A numeric value which is the sum of the values on the diagonal.

Note

If the argument x is not numeric, the function presents and error message and terminates. If the argument x is not a square matrix, the function presents an error message and terminates.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Examples

A <- matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE )
matrix.trace( A )

Maximum norm of matrix

Description

This function returns the max norm of a real matrix.

Usage

maximum.norm(x)

Arguments

Details

Let {\bf{x}} be an m \times n real matrix. The max norm returned is \left\| {\bf{x}} \right\|_{\max } = \mathop {\max }\limits_{i,j} \left| {x_{i,j} } \right|.

Value

A numeric value.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Golub, G. H. and C. F. Van Loan (1996). Matrix Computations, Third Edition, The John Hopkins University Press.

Horn, R. A. and C. R. Johnson (1985). Matrix Analysis, Cambridge University Press.

See Also

inf.norm, one.norm

Examples

A <- matrix( c( 3, 5, 7, 2, 6, 4, 0, 2, 8 ), nrow=3, ncol=3, byrow=TRUE )
maximum.norm( A )

Compute the one norm of a matrix

Description

This function returns the \left\| {\bf{x}} \right\|_1 norm of the matrix {\mathbf{x}}.

Usage

one.norm(x)

Arguments

Details

Let {\bf{x}} be an m \times n matrix. The formula used to compute the norm is \left\| {\bf{x}} \right\|_1 = \mathop {\max }\limits_{1 \le j \le n} \sum\limits_{i = 1}^m {\left| {x_{i,j} } \right|} . This is merely the maximum absolute column sum of the m \times n maxtris.

Value

A numeric value.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Golub, G. H. and C. F. Van Loan (1996). Matrix Computations, Third Edition, The John Hopkins University Press.

Horn, R. A. and C. R. Johnson (1985). Matrix Analysis, Cambridge University Press.

See Also

inf.norm

Examples

A <- matrix( c( 3, 5, 7, 2, 6, 4, 0, 2, 8 ), nrow=3, ncol=3, byrow=TRUE )
one.norm( A )

Pascal matrix

Description

This function returns an n by n Pascal matrix.

Usage

pascal.matrix(n)

Arguments

Details

In mathematics, particularly matrix theory and combinatorics, the Pascal matrix is a lower triangular matrix with binomial coefficients in the rows. It is easily obtained by performing an LU decomposition on the symmetric Pascal matrix of the same order and returning the lower triangular matrix.

Value

An order n matrix.

Note

If the argument n is not a positive integer, the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Aceto, L. and D. Trigiante, (2001). Matrices of Pascal and Other Greats, American Mathematical Monthly, March 2001, 232-245.

Call, G. S. and D. J. Velleman, (1993). Pascal's matrices, American Mathematical Monthly, April 1993, 100, 372-376.

Edelman, A. and G. Strang, (2004). Pascal Matrices, American Mathematical Monthly, 111(3), 361-385.

See Also

lu.decomposition, symmetric.pascal.matrix

Examples

P <- pascal.matrix( 4 )
print( P )

Store matrix inside another matrix

Description

This function returns a matrix which is a copy of matrix x into which the contents of matrix y have been inserted at the given row and column.

Usage

set.submatrix(x, y, row, col)

Arguments

Value

A matrix.

Note

If the argument x is not a numeric matrix, then the function presents an error message and stops. If the argument y is not a numeric matrix, then the function presents an error message and stops. If the argument row is not a positive integer, then the function presents an error message and stops. If the argument col is not a positive integer, then the function presents an error message and stops. If the target row range does not overlap with the row range of argument x, then the function presents an error message and stops. If the target col range does not overlap with the col range of argument x, then the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

Examples

x <- matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE )
y <- matrix( seq( 1, 4, 1 ), nrow=2, byrow=TRUE )
z <- set.submatrix( x, y, 3, 3 )

Shift matrix m rows down

Description

This function returns a matrix that has had its rows shifted downwards filling the above rows with the given fill value.

Usage

shift.down(A, rows = 1, fill = 0)

Arguments

Value

A matrix.

Note

If the argument A is not a numeric matrix, then the function presents an error message and stops. If the argument rows is not a positive integer, then the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

Examples

A <- matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE )
shift.down( A, 1 )
shift.down( A, 3 )

Shift a matrix n columns to the left

Description

This function returns a matrix that has been shifted n columns to the left filling the subsqeuent columns with the given fill value

Usage

shift.left(A, cols = 1, fill = 0)

Arguments

Value

A matrix.

Note

If the argument A is not a numeric matrix, then the function presents an error message and stops. If the argument cols is not a positive integer, then the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

Examples

A <- matrix( seq( 1, 12, 1 ), nrow=3, byrow=TRUE )
shift.left( A, 1 )
shift.left( A, 2 )

Shift matrix n columns to the right

Description

This function returns a matrix that has been shifted to the right n columns filling the previous columns with the given fill value.

Usage

shift.right(A, cols = 1, fill = 0)

Arguments

Value

A matrix.

Note

If the argument A is not a numeric matrix, then the function presents an error message and stops. If the argument rows is not a positive integer, then the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

Examples

A <- matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE )
shift.right( A, 1 )
shift.right( A, 2 )

Shift matrix m rows up

Description

This function returns a matrix where the argument as been shifted up the given number of rows filling the bottom rows with the given fill value.

Usage

shift.up(A, rows = 1, fill = 0)

Arguments

Value

A matrix.

Note

If the argument A is not a numeric matrix, then the function presents an error message and stops. If the argument rows is not a positive integer, then the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

Examples

A <- matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE )
shift.up( A, 1 )
shift.up( A, 3 )

Spectral norm of matrix

Description

This function returns the spectral norm of a real matrix.

Usage

spectral.norm(x)

Arguments

Details

Let {\bf{x}} be an m \times n real matrix. The function computes the order n square matrixmatrix {\bf{A}} = {\bf{x'}}\;{\bf{x}}. The R function eigen is applied to this matrix to obtain the vector of eigenvalues {\bf{\lambda }} = \left\lbrack {\begin{array}{cccc} {\lambda _1 } & {\lambda _2 } & \cdots & {\lambda _n } \\ \end{array}} \right\rbrack. By construction the eigenvalues are in descending order of value so that the largest eigenvalue is \lambda _1. Then the spectral norm is \left\| {\bf{x}} \right\|_2 = \sqrt {\lambda _1 }. If {\bf{x}} is a vector, then {\bf{L}}_2 = \sqrt {\bf{A}} is returned.

Value

A numeric value.

Note

If the argument x is not numeric, an error message is displayed and the function terminates. If the argument is neither a matrix nor a vector, an error message is displayed and the function terminates. If the product matrix {\bf{x'}}\;{\bf{x}} is negative definite, an error message displayed and the function terminates.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Golub, G. H. and C. F. Van Loan (1996). Matrix Computations, Third Edition, The John Hopkins University Press.

Horn, R. A. and C. R. Johnson (1985). Matrix Analysis, Cambridge University Press.

Examples

x <- matrix( c( 2, 4, 2, 1, 3, 1, 5, 2, 1, 2, 3, 3 ), nrow=3, ncol=4, byrow=TRUE )
spectral.norm( x )

Stirling Matrix

Description

This function constructs and returns a Stirling matrix which is a lower triangular matrix containing the Stirling numbers of the second kind.

Usage

stirling.matrix(n)

Arguments

Details

The Stirling numbers of the second kind, S_i^j, are used in combinatorics to compute the number of ways a set of i objects can be partitioned into j non-empty subsets j \le i. The numbers are also denoted by \left\{ {\begin{array}{c}i\\j\end{array}} \right\}. Stirling numbers of the second kind can be computed recursively with the equation S_j^{i + 1} = S_{j - 1}^i + j\;S_j^i,\quad 1 \le i \le n - 1,\;1 \le j \le i. The initial conditions for the recursion are S_i^i = 1,\quad 0 \le i \le n and S_j^0 = S_0^j = 0,\quad 0 \le j \le n. The resultant numbers are organized in an order n + 1 matrix \left\lbrack {\begin{array}{ccccc} {S_0^0}&0&0& \cdots &0\\ 0&{S_1^1}&0& \cdots &0\\ 0&{S_1^2}&{S_2^2}& \cdots &0\\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 0&{S_1^n}&{S_2^n}& \cdots &{S_n^n} \end{array}} \right\rbrack.

Value

An order n + 1 lower triangular matrix.

Note

If the argument n is not a positive integer, the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Aceto, L. and D. Trigiante (2001). Matrices of Pascal and Other Greats, American Mathematical Monthly, March 2001, 108(3), 232-245.

Examples

S <- stirling.matrix( 10 )
print( S )

SVD Inverse of a square matrix

Description

This function returns the inverse of a matrix using singular value decomposition. If the matrix is a square matrix, this should be equivalent to using the solve function. If the matrix is not a square matrix, then the result is the Moore-Penrose pseudo inverse.

Usage

svd.inverse(x)

Arguments

Value

A matrix.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

Examples

A <- matrix( c ( 1, 2, 2, 1 ), nrow=2, byrow=TRUE)
invA <- svd.inverse( A )
print( A )
print( invA )
print( A %*% invA )
B <- matrix( c( -1, 2, 2 ), nrow=1, byrow=TRUE )
invB <- svd.inverse( B )
print( B )
print( invB )
print( B %*% invB )

Symmetric Pascal matrix

Description

This function returns an n by n symmetric Pascal matrix.

Usage

symmetric.pascal.matrix(n)

Arguments

Details

In mathematics, particularly matrix theory and combinatorics, the symmetric Pascal matrix is a square matrix from which you can derive binomial coefficients. The matrix is an order n symmetric matrix with typical element given by {S_{i,j}} = {{n!} \mathord{\left/ {\vphantom {{n!} {\left[ {r!\;\left( {n - r} \right)!} \right]}}} \right. } {\left[ {r!\;\left( {n - r} \right)!} \right]}} where n = i + j - 2 and r = i - 1. The binomial coefficients are elegantly recovered from the symmetric Pascal matrix by performing an LU decomposition as {\bf{S}} = {\bf{L}}\;{\bf{U}}.

Value

An order n matrix.

Note

If the argument n is not a positive integer, the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Call, G. S. and D. J. Velleman, (1993). Pascal's matrices, American Mathematical Monthly, April 1993, 100, 372-376.

Edelman, A. and G. Strang, (2004). Pascal Matrices, American Mathematical Monthly, 111(3), 361-385.

Examples

S <- symmetric.pascal.matrix( 4 )
print( S )

Toeplitz Matrix

Description

This function constructs an order n Toeplitz matrix from the values in the order 2 * n - 1 vector x.

Usage

toeplitz.matrix(n, x)

Arguments

Details

The element T[i,j] in the Toeplitz matrix is x[i-j+n].

Value

An order n matrix.

Note

If the argument n is not a positive integer, the function presents an error message and stops. If the length of x is not equal to 2 * n - 1, the function presents an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Monahan, J. F. (2011). Numerical Methods of Statistics, Cambridge University Press.

Examples

T <- toeplitz.matrix( 4, seq( 1, 7 ) )
print( T )

u vectors of an identity matrix

Description

This function constructs an order n * ( n + 1 ) / 2 identity matrix and an order matrix u that that maps the ordered pair of indices (i,j) i=j, ..., n; j=1, ..., n to a column in this identity matrix.

Usage

u.vectors(n)

Arguments

Details

The function firsts constructs an identity matrix of order \frac{1}{2}n\left( {n + 1} \right). {{{\bf{u}}_{i,j}}} is the column vector in the order \frac{1}{2}n\left( {n + 1} \right) identity matrix for column k = \left( {j - 1} \right)n + i - \frac{1}{2}j\left( {j - 1} \right).

Value

A list with two named components

Note

If the argument is not an integer, the function displays an error message and stops. If the argument is less than two, the function displays an error message and stops.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1980). The elimination matrix, some lemmas and applications, SIAM Journal on Algebraic Discrete Methods, 1(4), December 1980, 422-449.

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

Examples

u <- u.vectors( 3 )

Upper triangle portion of a matrix

Description

Returns the lower triangle including the diagonal of a square numeric matrix.

Usage

upper.triangle(x)

Arguments

Value

A matrix.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

See Also

is.square.matrix

Examples

A <- matrix( seq( 1, 9, 1 ), nrow=3, byrow=TRUE )
upper.triangle( A )

Vandermonde matrix

Description

This function returns an m by n matrix of the powers of the alpha vector

Usage

vandermonde.matrix(alpha, n)

Arguments

Details

In linear algebra, a Vandermonde matrix is an m \times n matrix with terms of a geometric progression of an m \times 1 parameter vector {\bf{\alpha }} = {\left\lbrack {\begin{array}{cccc} {{\alpha _1}}&{{\alpha _2}}& \cdots &{{\alpha _m}} \end{array}} \right\rbrack^\prime }

such that V\left( {\bf{\alpha }} \right) = \left\lbrack {\begin{array}{ccccc} 1&{{\alpha _1}}&{\alpha _1^2}& \cdots &{\alpha _1^{n - 1}}\\ 1&{{\alpha _2}}&{\alpha _2^2}& \cdots &{\alpha _2^{n - 1}}\\ 1&{{\alpha _3}}&{\alpha _3^2}& \cdots &{\alpha _3^{n - 1}}\\ \cdots & \cdots & \cdots & \cdots & \cdots \\ 1&{{\alpha _m}}&{\alpha _m^2}& \cdots &{\alpha _m^{n - 1}} \end{array}} \right\rbrack.

Value

A matrix.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Horn, R. A. and C. R. Johnson (1991). Topics in matrix analysis, Cambridge University Press.

Examples

alpha <- c( .1, .2, .3, .4 )
V <- vandermonde.matrix( alpha, 4 )
print( V )

Vectorize a matrix

Description

This function returns a column vector that is a stack of the columns of x, an m by n matrix.

Usage

vec(x)

Arguments

Value

A matrix with m\;n rows and one column.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

Examples

x <- matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE )
print( x )
vecx <- vec( x )
print( vecx )

Vectorize a matrix

Description

This function returns a stack of the lower triangular matrix of a square matrix as a matrix with 1 column and n * ( n + 1 ) / 2 rows

Usage

vech(x)

Arguments

Value

A matrix with \frac{1}{2}n\left( {n + 1} \right) rows and one column.

Author(s)

Frederick Novomestky fnovomes@poly.edu

References

Magnus, J. R. and H. Neudecker (1999) Matrix Differential Calculus with Applications in Statistics and Econometrics, Second Edition, John Wiley.

See Also

is.square.matrix

Examples

x <- matrix( seq( 1, 16, 1 ), nrow=4, byrow=TRUE )
print( x )
y <- vech( x )
print( y )