Polynomial in a Matrix

Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Computer Science #◇Vector Spaces #◇Linear Transformations

◇Overview

Definition: If A is an n×n square matrix and p(x) is a polynomial defined by:

◇Details / Theory

Definition: If A is an n×n square matrix and p(x) is a polynomial defined by:
p(x)=ckxk+ck−1xk−1+⋯+c1x+c0
where c0,c1,…,ck are scalars, then p(A) is the n×n matrix defined as:
p(A)=ckAk+ck−1Ak−1+⋯+c1A+c0In
(Handout 2, Page 9)
Personal Note: The constant term c0 becomes c0In when evaluating the polynomial for a matrix. Example: If p(x)=2x2−3x+4 and A=[10 23].
Then p(A)=2A2−3A+4I2.
First, calculate A2:
A2=A⋅A=[10 23][10 23]=[1(1)+0(2)1(0)+0(3) 2(1)+3(2)2(0)+3(3)]=[10 89]
Then,
p(A)=2[10 89]−3[10 23]+4[10 01]p(A)=[20 1618]−[30 69]+[40 04]=[2−3+40−0+0 16−6+018−9+4]=[30 1013]
(Example based on Handout 2, Page 9)

◇Common Pitfalls

Matrix polynomials require the matrix to be square (powers only defined for square matrices). The Cayley-Hamilton theorem states $p(A) = O$ for the characteristic polynomial — don’t confuse this with $p(\lambda) = 0$ .

◇Summary

TL;DR — A matrix polynomial $p(A) = c_n A^n + \cdots + c_1 A + c_0 I$ substitutes a square matrix into a polynomial. Every matrix satisfies its own characteristic polynomial (Cayley-Hamilton).

◇Links

Related: Source / Reference:

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Related Notes: ◇Matrix Multiplication, ◇Elementary Row Operation, ◇Matrices Detailed

POLYNOMIAL IN A MATRIX