18.4 Functions of a Matrix

: expm (A)

Return the exponential of a matrix.

The matrix exponential is defined as the infinite Taylor series

expm (A) = I + A + A^2/2! + A^3/3! + …

However, the Taylor series is not the way to compute the matrix exponential; see Moler and Van Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Review, 1978. This routine uses Ward’s diagonal Padé approximation method with three step preconditioning (SIAM Journal on Numerical Analysis, 1977). Diagonal Padé approximations are rational polynomials of matrices

-1
D (A)   N (A)

whose Taylor series matches the first 2q+1 terms of the Taylor series above; direct evaluation of the Taylor series (with the same preconditioning steps) may be desirable in lieu of the Padé approximation when Dq(A) is ill-conditioned.

See also: logm, sqrtm.

: s = logm (A)

: s = logm (A, opt_iters)

: [s, iters] = logm (…)

Compute the matrix logarithm of the square matrix A.

The implementation utilizes a Padé approximant and the identity

logm (A) = 2^k * logm (A^(1 / 2^k))

The optional input opt_iters is the maximum number of square roots to compute and defaults to 100.

The optional output iters is the number of square roots actually computed.

See also: expm, sqrtm.

: s = sqrtm (A)

: [s, error_estimate] = sqrtm (A)

Compute the matrix square root of the square matrix A.

Ref: N.J. Higham. A New sqrtm for MATLAB. Numerical Analysis Report No. 336, Manchester Centre for Computational Mathematics, Manchester, England, January 1999.

See also: expm, logm.

: kron (A, B)

: kron (A1, A2, …)

Form the Kronecker product of two or more matrices.

This is defined block by block as

x = [ a(i,j)*b ]

For example:

kron (1:4, ones (3, 1))
     ⇒  1  2  3  4
         1  2  3  4
         1  2  3  4

If there are more than two input arguments A1, A2, …, An the Kronecker product is computed as

kron (kron (A1, A2), …, An)

Since the Kronecker product is associative, this is well-defined.

: blkmm (A, B)

Compute products of matrix blocks.

The blocks are given as 2-dimensional subarrays of the arrays A, B. The size of A must have the form [m,k,…] and size of B must be [k,n,…]. The result is then of size [m,n,…] and is computed as follows:

for i = 1:prod (size (A)(3:end))
  C(:,:,i) = A(:,:,i) * B(:,:,i)
endfor

: X = sylvester (A, B, C)

Solve the Sylvester equation.

The Sylvester equation is defined as:

A X + X B = C

The solution is computed using standard LAPACK subroutines.

For example:

sylvester ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12])
   ⇒ [ 0.50000, 0.66667; 0.66667, 0.50000 ]