|MATLAB Function Reference|
Matrix polynomial evaluation
Y = polyvalm(p,X)
evaluates a polynomial in a matrix sense. This is the same as substituting matrix
X in the 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.
Its characteristic polynomial can be generated with the
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.
But evaluating it in a matrix sense is interesting.
The result is the zero matrix. This is an instance of the Cayley-Hamilton theorem: a matrix satisfies its own characteristic equation.