7.3 PROPERTIES OF THE CHARACTERISTIC POLYNOMIAL

The characteristic equation of an n n square matrix, A, can be written as,

and if the equation has n distinct roots that are the eigenvalues of A, the polynomial may be written in factored form

If ? is set equal to zero, the representations of eqs (7.8) provide

which shows that the product of the eigenvalues is equal to the determinant of the matrix A. Indeed, if A is nonsingular, A cannot posess a zero eigenvalue and, conversely, if just one eigenvalue of A is zero, then A must singular.

Suppose n = 4. If det K = det[ A ? ? I] is expanded, or if the factors of eq (7.8b) are multiplied, then

A comparison to eq (7.8a) along with some mathmematical induction indicates that ? _{n ?1}, the coefficient of the ( n ?1) ^th term, is

and an expanasion of det[ A ? ? I] can also be conducted to show that the coefficient of the (n ?1) ^th term is,

The comparison of these two expressions for ? _{n ?1} shows that the sum of the eigenvalues of A is equal to the trace of A.