The MatrixFunction(A) command returns the Matrix obtained by interpolating [lambda, F( lambda )] for each of the eigenvalues lambda of A, including multiplicities. Here the Matrix polynomial is r(lambda) = F(lambda) - p(lambda)*q(lambda) where p(x) is the characteristic polynomial, q(lambda) is the quotient, and r(lambda) is the remainder.

The options option provides additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix constructor that builds the result. These options may also be provided in the form outputoptions=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions option, the latter takes precedence (regardless of the order).

This function is part of the LinearAlgebra package, and so it can be used in the form MatrixFunction(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[MatrixFunction](..).

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[-13\,-10\right]\,\left[21\,16\right]\right]\right)$

$A≔\left[\begin{array}{cc}\mathrm{-13}& \mathrm{-10}\\ 21& 16\end{array}\right]$

$\mathrm{MatrixFunction}\left(A\,v^2\,v\right)$

$\left[\begin{array}{cc}\mathrm{-41}& \mathrm{-30}\\ 63& 46\end{array}\right]$

$\mathrm{MatrixFunction}\left(A\,\sin \left(x\right)\,x\right)$

$\left[\begin{array}{cc}15\sin \left(1\right)-14\sin \left(2\right)& -10\sin \left(2\right)+10\sin \left(1\right)\\ 21\sin \left(2\right)-21\sin \left(1\right)& -14\sin \left(1\right)+15\sin \left(2\right)\end{array}\right]$

$\mathrm{MatrixFunction}\left(A\,\exp \left(x\right)\,x\right)$

$\left[\begin{array}{cc}15\ⅇ-14{\ⅇ}^2& -10{\ⅇ}^2+10\ⅇ\\ 21{\ⅇ}^2-21\ⅇ& -14\ⅇ+15{\ⅇ}^2\end{array}\right]$

$\mathrm{MatrixExponential}\left(A\,t\right)$

$\left[\begin{array}{cc}15{\ⅇ}^t-14{\ⅇ}^{2t}& -10{\ⅇ}^{2t}+10{\ⅇ}^t\\ 21{\ⅇ}^{2t}-21{\ⅇ}^t& -14{\ⅇ}^t+15{\ⅇ}^{2t}\end{array}\right]$

$\mathrm{MatrixFunction}\left(A\,\exp \left(xt\right)\,x\right)$

$\left[\begin{array}{cc}15{\ⅇ}^t-14{\ⅇ}^{2t}& -10{\ⅇ}^{2t}+10{\ⅇ}^t\\ 21{\ⅇ}^{2t}-21{\ⅇ}^t& -14{\ⅇ}^t+15{\ⅇ}^{2t}\end{array}\right]$

$\mathrm{MatrixFunction}\left(A\,\ln \left(x\right)\,x\,\mathrm{readonly}\right)$

$\left[\begin{array}{cc}-14\ln \left(2\right)& -10\ln \left(2\right)\\ 21\ln \left(2\right)& 15\ln \left(2\right)\end{array}\right]$

$B≔\mathrm{Matrix}\left(\left[\left[y\right]\right]\right)$

$B≔\left[\begin{array}{c}y\end{array}\right]$

$\mathrm{MatrixExponential}\left(B\right)$

$\left[\begin{array}{c}{\ⅇ}^y\end{array}\right]$

$\mathrm{MatrixFunction}\left(B\,\exp \left(y\right)\,y\right)$

$\left[\begin{array}{c}{\ⅇ}^y\end{array}\right]$

$\mathrm{MatrixFunction}\left(B\,\exp \left(y\right)\right)$

$\left[\begin{array}{c}{\ⅇ}^y\end{array}\right]$