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# polyvalm

Matrix polynomial evaluation

## Syntax

Y = polyvalm(p,X)

## Description

Y = polyvalm(p,X) evaluates a polynomial in a matrix sense. This is the same as substituting matrix X in the polynomial p.

Polynomial p is a vector whose elements are the coefficients of a polynomial in descending powers, and X must be a square matrix.

## Examples

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

```X = pascal(4)
X =
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(X)
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 is not very interesting.

``` polyval(p,X)
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,X)
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 satisfies its own characteristic equation.

## See Also

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