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 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).)
Read more about this topic: Characteristic Polynomial
Famous quotes containing the words formal and/or definition:
“The spiritual kinship between Lincoln and Whitman was founded upon their Americanism, their essential Westernism. Whitman had grown up without much formal education; Lincoln had scarcely any education. One had become the notable poet of the day; one the orator of the Gettsyburg Address. It was inevitable that Whitman as a poet should turn with a feeling of kinship to Lincoln, and even without any association or contact feel that Lincoln was his.”
—Edgar Lee Masters (1869–1950)
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)