Characteristic polynomial

In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace.

Table of contents

1 Motivation 2 Intuitive content 3 Formal definition 4 Properties

Motivation

In the case of a diagonal matrix, the characteristic polynomial is easy to define: if the diagonal entries are a, b, c the characteristic polynomial will be (t-a)(t-b)(t-c)... up to a convention about sign (+ or -). That is, the diagonal entries become the rootss of the characteristic polynomial. This is not really enough to explain the definition in the general case. But if we add the condition that similar matrices A and B^-1AB should have the same characteristic polynomial, it essentially forces the definition given later. If M and N are similar matrices, then they also have the same characteristic polynomial. The converse however is not true: matrices with the same characteristic polynomial need not be similar.

Intuitive content

The geometric reasons that can be given for the statements just made are these. Every square matrix M is as close as we like to a matrix M* that is similar to a diagonal matrix. Therefore, assuming continuity, everything is forced by the definition working up to similarity. On the other hand, we can't assume that 'similarity up to the limit implies similarity at the limit': the transformation we use can itself go out of control in a limiting process.

Formal definition

We start with a field K (you can think of K as the real or complex numbers) and an n-by-n matrix A over K. The characteristic polynomial of A, denoted by p_A(t), is the element of the polynomial ring K[t] defined by

p_A(t) = det(A - tI)

where I denotes the n-by-n identity matrix. This is indeed a polynomial, since determinants are defined in terms of sums of products. (Some authors define the characteristic polynomial to be det(tI - A); the difference is immaterial since the two polynomials differ at most by a sign.)

Properties

The degree of the polynomial p_A(t) is n. The most important fact about the characteristic polynomial is this: the eigenvalues of A are precisely the zeros of p_A(t). The constant coefficient p_A(0) is equal to the determinant of A, and the coefficient of t^n-1 is equal to (-1)^n-1 times the trace of A.

For 2×2 matrices, the characteristic polynomial of A is nicely expressed then as

t²-tr(A)t+det(A)

where tr(A) represents the matrix trace of A and det(A) the determinant of A.

The Cayley-Hamilton theorem states that replacing t by A in the expression for p_A(t) yields the zero matrix: p_A(A) = 0. Simply, every matrix satisfies its own characteristic equation. As a consequence of this, one can show that the minimal polynomial of A divides the characteristic polynomial of A.

The matrix A and its transpose have the same characteristic polynomial. A is similar to a triangular matrix if and only if its characteristic polynomial can be completely factored into linear factors over K. In fact, A is even similar to a matrix in Jordan normal form in this case.