Characteristic polynomialIn 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.
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 pA(t), is the element of the polynomial ring K[t] defined by
- pA(t) = det(A - tI)
The degree of the polynomial pA(t) is n. The most important fact about the characteristic polynomial is this: the eigenvalues of A are precisely the zeros of pA(t). The constant coefficient pA(0) is equal to the determinant of A, and the coefficient of tn-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
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.
The matrix A and its transpose have the same characteristic polynomial. If A and B 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.
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.