Matrix polynomial evaluation

`Y = polyvalm(p,X)`

`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.

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.

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