Y = polyvalm(p,S)
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
Its characteristic polynomial can be generated with theS = pascal(4)
S =
1 1 1 1
1 2 3 4
1 3 6 10
1 4 10 20
poly
function.
This represents the polynomialp = poly(S)
p =
1 -29 72 -29 1
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.
But evaluating it in a matrix sense is interesting.polyval(p,S)
ans =
16 16 16 16
16 15 -140 -563
16 -140 -2549 -12089
16 -563 -12089 -43779
The result is the zero matrix. This is an instance of the Cayley-Hamilton theorem: a matrix that satisfies its own characteristic equation.polyvalm(p,S)
ans =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
conv
,poly
,polyval
,residue
,roots
(c) Copyright 1994 by The MathWorks, Inc.