Both the matrix and the polynomial coefficients can be symbolic:

Matrix polynomial evaluation of a real matrix:

Matrix polynomial evaluation of a complex matrix:

Matrix polynomial evaluation of an exact matrix:

Compute a polynomial function—expressed as a list of coefficients—of a symbolic matrix:

Recompute with the polynomial expressed in explicit form:

Verify that both results are consistent with the definition:

Evaluate the polynomial of spare matrices; the result is also sparse:

Format the result:

Evaluate the polynomial of structured matrices including symmetrized arrays:

For quantity arrays, the coefficients must have units for the sum to be dimensionally consistent:

Use MatrixPolynomialValue with an identity matrix:

The result is equivalent to multiplying the identical matrix by the polynomial evaluated at the point :

Use MatrixPolynomialValue with a diagonal matrix:

The result is equivalent to evaluating the polynomial at each of the diagonal entries:

The polynomial value of JordanMatrix[λ,k] yields the polynomial on the diagonal plus a correction above the diagonal:

Each additional power involves another superdiagonal: