Engineering Analysis/Matrix Functions

If we have functions, and we use a matrix as the input to those functions, the output values are not always intuitive. For instance, if we have a function f(x), and as the input argument we use matrix A, the output matrix is not necessarily the function f applied to the individual elements of A.

Diagonal Matrix

In the special case of diagonal matrices, the result of f(A) is the function applied to each element of the diagonal matrix:

$A = \begin{bmatrix} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{nn} \end{bmatrix}$

Then the function f(A) is given by:

$f(A) = \begin{bmatrix} f(a_{11}) & 0 & \cdots & 0 \\ 0 & f(a_{22}) & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & f(a_{nn}) \end{bmatrix}$

Jordan Cannonical Form

Matrices in Jordan Canonical form also have an easy way to compute the functions of the matrix. However, this method is not nearly as easy as the diagonal matrices described above.

If we have a matrix in Jordan Block form, A, the function f(A) is given by:

$f(A) = \begin{bmatrix} \frac{f(a)}{0!} & \frac{f'(a)}{1!} & \cdots & \frac{f^{(r-1)}(a)}{(r-1)!} \\ 0 & \frac{f(a)}{0!} & \cdots & \frac{f^(r-2)(a)}{(r-2)!} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \frac{f(a)}{0!} \end{bmatrix}$

The matrix indices have been removed, because in Jordan block form, all the diagonal elements must be equal.

If the matrix is in Jordan Block form, the value of the function is given as the function applied to the individual diagonal blocks.