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.
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: