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.

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 Av = λv, or (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(λI - A) are the eigenvalues of A. Since this function is a polynomial in λ, 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_by_n matrix A over K. The characteristic polynomial of A, denoted by p_A(t), is the polynomial defined by

p_A(t) = det(tI - 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 - tI); 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

We have to compute the determinant of

and this determinant is

(t - 2)t - 1( - 1) = t² - 2t + 1

The latter is the characteristic polynomial of A.

Properties

The polynomial p_A(t) is monic (its highest 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 the trace of A.

For a 2×2 matrix A, the characteristic polynomial 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.

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.