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Functions of Matrices

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This Demonstration computes some standard functions of a set of rather arbitrary matrices. The test matrix has distinct eigenvalues; the matrices and are symbolic, but triangular with different and multiple eigenvalues; the matrices to are numeric with the same multiple eigenvalues but different Jordan decomposition forms; is a numerical random matrix.

Contributed by: Mikhail Dimitrov Mikhailov (March 2011)
Open content licensed under CC BY-NC-SA

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Different methods of computing a function of a matrix are described in: F. R. Gantmacher, The Theory of Matrices, trans. K. A. Hirsch, 2 vols., New York: Chelsea Publishing Company, 1959.

This Demonstration uses the matrix exponential of a matrix with no zero eigenvalues to compute an arbitrary function of the matrix. Replacing by sin, cos, , , , , or erf computes the corresponding function of the matrix.

The matrix satisfies the matrix differential equation:

if or ,

if or ,

if or ,

if or ,

if .

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