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.
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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,
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(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 pA(t ), is the polynomial defined by
- pA(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
- <math>A=\begin{pmatrix}
2 & 1\\ -1& 0 \end{pmatrix}. <math> We have to compute the determinant of
- <math>t I-A = \begin{pmatrix}
t-2&-1\\ 1&t \end{pmatrix} <math> and this determinant is
- <math>(t-2)t - 1(-1) = t^2-2t+1.\,\!<math>
The latter is the characteristic polynomial of A.
Properties
The polynomial pA(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 pA(t ). The constant coefficient pA(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
- t 2 - 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 pA(t ) yields the zero matrix: pA(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.de:Charakteristisches Polynom