Characteristic polynomial

of a matrix over a field

The polynomial

over . The degree of the characteristic polynomial is equal to the order of the square matrix , the coefficient is the trace of (, cf. Trace of a square matrix), the coefficient is the sum of all principal minors of order , in particular, (cf. Minor). The equation is called the characteristic equation of or the secular equation.

The roots of the characteristic polynomial lying in are called the characteristic values or eigen values of . If is a number field, then the term "characteristic number of a matrixcharacteristic numbers" is also used. Sometimes the roots of the characteristic polynomial are considered in the algebraic closure of . They are usually called the characteristic roots of . A matrix of order regarded over an algebraically closed field (for example, over the field of complex numbers) has eigen values , if every root is counted according to its multiplicity. See also Eigen value.

Similar matrices have the same characteristic polynomial. Every polynomial over with leading coefficient is the characteristic polynomial of some matrix over of order , the so-called Frobenius matrix.

For references see Matrix.

Comments

The characteristic roots are often also called the eigen values or characteristic values, thereby not distinguishing the roots of the characteristic polynomial in and those in its algebraic closure. Given a polynomial . The matrix in companion form

with , has as its characteristic polynomial.