Characteristic Polynomial - Formal Definition

Formal Definition

We start with a field K (such 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 and the determinant is being taken in K, the ring of polynomials in t over K. (Some authors define the characteristic polynomial to be det(A − t I). That polynomial differs from the one defined here by a sign (−1)n, so it makes no difference for properties like having as roots the eigenvalues of A; however the current definition always gives a monic polynomial, whereas the alternative definition always has constant term det(A).)