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.