MATLAB Function Reference
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.

`polyfit`, `polyval`