6. Polynomials and functions of matrices

and A = (a_ij )_n x n . Define

If A is diagonalizable, A = TDT ^{- 1} where D = diag (₁ , ..., _n ), then

If p is the characteristic polynomial of A, then p(D) = 0, because

D^k = diag (₁ ^k, ..., _n ^k)    , k = 0, 1, ..., n

=>      p(D) =		p(1 )		0		= 0n x n
			·.
		0		p(n )

It now follows from (1) that p(A) = 0 (when A is diagonalizable). More generally,

the Cayley-Hamiltonin theorem: If p is the characteristic polynomail of a square matrix A ,then p (A) = 0, that is, A satisfies its own characteristic equation.

Example 1: Cayley-Hamilton theorem

Let p be the characteristic polynomial of A.

(2)

p() = ( - 1)ⁿⁿ + b_{n - 1} ^{n - 1} + ··· + b₁ + b₀ = 0

(3)

where we can substitute Aⁿ from the second last equation. It follows that all the powers A^k, k n, can be written in terms of A⁰, A¹, ..., A^n-1.

Example 2: Powers of a matrix

If A ^{- 1} exists, that is, if = 0 is not an eigenvalue of A => det A = p(0) = b₀ 0, it follows from (3) that

and thus

(4)

Next consider functions f (A) of matrices, where f : C -> C can be expressed as a power series

(5)

f(z) = c_k z^k.

We have already mentioned the functions e^A, sin A and cos A before. (Section 5.1 and exercises) Define

(6)

f(A) = c_k A^k         (A⁰ = I)

if the series converges. The following result can be shown:

Proposition 1. If the radius of convergence of the power series (5) is r ( f can be expressed as (5) when z, | z | < r ), then the matrix series (6) converges if the spectral radius r (A) of A satisfies r (A) < r. (r(A) = max{| ₁ | , ..., | _n |}).

Since all the powers A^k, k n, can be written in terms of A⁰, A¹, ..., A^n-1, it follows from (6) that

(7)

The coefficients d₀ , ..., d_{n - 1} depend from A. They satisfy the equation

f(_i ) = d₀ + d_{1 i} + d₂ ²_i + ··· + d_{n - 1 i} ^{n - 1},    i = 1, ..., n

where _i 's are the eigenvalues of A.

f(_i ) = c_{k i} ^k  , p (_i ) = 0   =>    f(_i ) = d₀ + d_{1 i} + ··· + d_{n - 1 i} ^{n - 1}

Example 3: sin A

Note. If q() = 0, where q is a polynomial, it is not necessarily true that q(A) = 0. There exists a polynomial of minimal degree, called the minimal polynomial of A, such that m(A) = 0.

If the multiplicity of an eigenvalue _i is bigger than one, we may also use the equations

f '(_i ) = d₁ + 2d_{2 i} + ··· + (n - 1)d_{n - 1 i} ^{n - 2}

If A is diagonalizable, then

f(A)	= ck Ak = ck TDkT - 1 = T [ ck diag (i k, ..., n k) ] T - 1
	= T [ diag (ck i k, ..., ck n k) ] T -1
	= T diag (ck 1 k, ..., ck n k)T - 1
	= T diag (f (1 ), ..., f (n ))T - 1

and thus

(8)

f(A) = T diag (f (₁ ), ..., f (_n ))T ^{- 1}.