# polyvalm

## Purpose

Matrix polynomial evaluation.

## Synopsis

    Y = polyvalm(p,S)

## Description

`polyvalm(p,S)` evaluates a polynomial in a matrix sense. `p` is a vector whose elements are the coefficients of a polynomial in descending powers, and `S` is a square matrix

## Examples

The Pascal matrices are formed from Pascal's triangle of binomial coefficients. Here is the Pascal matrix of order 4.

     S = pascal(4)

     S =
        1    1    1    1
        1    2    3    4
        1    3    6   10
        1    4   10   20

Its characteristic polynomial can be generated with the `poly` function.

     p = poly(S)

     p =
        1    -29    72    -29    1

This represents the polynomial

Pascal matrices have the curious property that the vector of coefficients of the characteristic polynomial is palindromic - it is the same forward and backward.

Evaluating this polynomial at each element of `S` is not very interesting.

     polyval(p,S)

     ans =
        16      16      16      16
        16      15    -140    -563
        16    -140   -2549  -12089
        16    -563  -12089  -43779

But evaluating it in a matrix sense is interesting.

     polyvalm(p,S)

     ans =
        0    0    0    0
        0    0    0    0
        0    0    0    0
        0    0    0    0

The result is the zero matrix. This is an instance of the Cayley-Hamilton theorem: a matrix that satisfies its own characteristic equation.

## See Also

    conv, poly, polyval, residue, roots

(c) Copyright 1994 by The MathWorks, Inc.