Characteristic polynomial

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

Contents

1 Motivation
2 Formal definition
3 Example
4 Properties
5 Characteristic polynomial of a product of two matrices
6 See also

Motivation

Given a square matrix A, we want to find a polynomial whose roots are precisely the eigenvalues of A. For a diagonal matrix A, 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 −). This works because the diagonal entries are also the eigenvalues of this matrix.

For a general matrix A, one can proceed as follows. If λ is an eigenvalue of A, then there is an eigenvector v≠0 such that

A v = λv,

(λI − A)v = 0

(where I is the identity matrix). Since v is non-zero, this means that the matrix λI − A is singular, which in turn means that its determinant is 0. We have just shown that the roots of the function det(t I − A) are the eigenvalues of A. Since this function is a polynomial in t, we're done.

Formal definition

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

p_A(t) = det(t I − A)

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(A − t I); the difference is immaterial since the two polynomials differ at most by a sign.)

Example

Suppose we want to compute the characteristic polynomial of the matrix

[A=\begin2 & 1\\-1& 0\end.]

We have to compute the determinant of

[t I-A = \begint-2&-1\\1&t\end]

and this determinant is

[(t-2)t - 1(-1) = t^2-2t+1.\,\!]

The latter is the characteristic polynomial of A.

Properties

The polynomial p_A(t) is monic (its leading coefficient is 1) and its degree is n. The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph: the eigenvalues of A are precisely the roots of p_A(t). The constant coefficient p_A(0) is equal to (−1)ⁿ times the determinant of A, and the coefficient of t^{n − 1} is equal to the negative of tr(A), the matrix trace of A. For a 2×2 matrix A, the characteristic polynomial is nicely expressed then as

t² − tr(A)t + det(A).

All real polynomials of odd degree have a real number as a root, so for odd n, every real matrix has at least one real eigenvalue. Many real polynomials of even degree do not have a real root, but the fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of real polynomials, hence the non-real eigenvalues, come in conjugate pairs.

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.

Two similar matrices have the same characteristic polynomial. The converse however is not true in general: two matrices with the same characteristic polynomial need not be similar.

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.