Powers of a Matrix

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

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◇Overview

Definition: If A is a square matrix, non-negative integer powers of A are defined as:

◇Key Concepts

Definition Laws of Exponents Caution

◇Details / Theory

Definition: If A is a square matrix, non-negative integer powers of A are defined as:
A0=In​ (Identity matrix of the same order as A)
A1=A

Ak=A⋅A⋅…⋅A (k times), for k≥1.
(Handout 2, Page 8) Laws of Exponents: If A is a square matrix and r,s are non-negative integers:
ArAs=Ar+s

(Ar)s=Ars
(Handout 2, Page 8) Caution: In general, (AB)k=AkBk unless A and B commute (i.e., AB=BA).

◇Common Pitfalls

Only square matrices can be raised to powers ($A^k = \underbrace{A \cdot A \cdots A}_{k}$). $A^0 = I$ by convention (the identity matrix, not the number 1). $(AB)^k \neq A^k B^k$ unless $A$ and $B$ commute.

◇Summary

TL;DR — $A^k$ means multiplying $A$ by itself $k$ times. Only for square matrices. $A^0 = I$. Powers of diagonal matrices are computed entry-wise.

◇Links

Related: Source / Reference:

◇References & Navigation

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