TABLE OF CONTENTS The characteristic polynomial plays a central role in the theoretical development of linear algebra and matrix analysis, but it is not alone in this respect. There are other polynomials that occur naturally, and the purpose of this section is to explore some of them.
In this section it is convenient to consider the characteristic polynomial of A ? C n n to be c( x) = det ( x I ? A) . This differs from the definition given on p. 492 only in the sense that the coefficients of c( x) = det ( x I ? A) have different signs than the coefficients of ?( x) = det ( A ? x I) . In particular, c( x) is a monic polynomial (i.e., its leading coefficient is 1), whereas the leading coefficient of ?( x) is ( ?1) n . (Of course, the roots of c and ? are identical.)
Monic polynomials p( x) such that p( A) = 0 are said to be annihilating polynomials for A . For example, the Cayley-Hamilton theorem (pp. 509, 532) guarantees that c( x) is an annihilating polynomial of degree n.
There is a unique annihilating polynomial for A of minimal degree, and this polynomial, denoted by m(
