Matrix Polynomials, An Introduction

Introduction

Given a ring R, the n×n matrices over R can be added, scaled, and multiplied. If x is such a matrix, we can talk about x+3, 7x-11, or x³-5x²+x+21. In other words, a matrix can be substituted into a polynomial, just like a scalar from R.

The coefficients of the polynomial usually come from R; this will be our default assumption. If you want everything to be a matrix, change a coefficient, like 5, into 5 times the identity matrix. Now 5 can be multiplied by x² using matrix multiplication. It's really the same whether you turn 5 into a matrix or leave it as a scalar, as long as you know how to apply the action of 5 to x².