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 pA(t), is the polynomial defined by

pA(t) = det(t IA)

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(At 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).)

Read more about this topic:  Characteristic Polynomial

Famous quotes containing the words formal and/or definition:

    On every formal visit a child ought to be of the party, by way of provision for discourse.
    Jane Austen (1775–1817)

    ... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.
    Adrienne Rich (b. 1929)