We start with a field K (you can think of K as the real or complex numbers) and an nbyn 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
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^{n1} is equal to (1)^{n1} times the trace of A.
The CayleyHamilton 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. 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.
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