WOLFRAM NOTEBOOK

Functions of Matrices

This Demonstration computes some standard functions of a set of rather arbitrary matrices. The test matrix
A
has distinct eigenvalues; the matrices
B
1
and
B
2
are symbolic, but triangular with different and multiple eigenvalues; the matrices
K
1
to
K
8
are numeric with the same multiple eigenvalues but different Jordan decomposition forms;
R
is a numerical random matrix.

Details

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
f
of the matrix. Replacing
f
by sin, cos,
J
0
,
Y
0
,
I
0
,
K
0
, or erf computes the corresponding function of the matrix.
The matrix
Y(x)
satisfies the matrix differential equation:
Y''(x)=-M.M.Y(x)
if
Y(x)=sin(Mx)
or
Y(x)=cos(Mx)
,
Y''(x)=M.M.Y(x)
if
Y(x)=sinh(Mx)
or
Y(x)=cosh(Mx)
,
Y''(x)+Y'(x)/x=-M.M.Y(x)
if
Y(x)=
J
0
(Mx)
or
Y(x)=
Y
0
(Mx)
,
Y''(x)+Y'(x)/x=-M.M.Y(x)
if
Y(x)=
I
0
(Mx)
or
Y(x)=
K
0
(Mx)
,
Y''(x)=-2xM.M.Y'(x)
if
Y(x)=erfc(Mx)
.

Permanent Citation

Mikhail Dimitrov Mikhailov

​"Functions of Matrices"​
http://demonstrations.wolfram.com/FunctionsOfMatrices/
Wolfram Demonstrations Project
​Published: March 7, 2011
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