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The companion matrix T h e companion

matrix

C of any polynomial (E.l)

Pn (A) = A"" + a^-iA"""^ + . . . + ao is defined t o be the matrix 0 0 ... 0 1 0 ... 0 0 1 ... 0

-ao -ai -a2

00 . 00. From the point of view of proving Theorem 20 it suffices to show t h a t the eigenvalues Ai of C satisfy pn (Xi) = 0. To do this we note t h a t the eigenvalues of C are also those of C'^ and consider the equation C ^ x = xA. If x = [^r^], equating the first n — 1 elements of C ^ x and xA gives Xi^i = Xxi,

2 = 1, 2 , . . . , n — 1

while equating the final elements gives

I]^-

iXi

(E.2)

XXr,

i=l

This implies t h a t xi 7^ 0 (since if x i = 0 then x = 0 and cannot be an eigenvector), and t h a t Xi = X^~^xi, I < i < n. Substituting these values for Xi and Xn in equation (E.2) then gives n 1=1

or, from equation (E.l) and since xi 7^ 0, pn (A) = 0. Thus any eigenvalue A^ of C must satisfy p^ {Xi) — 0. Conversely, if p^ (A) = 0, then x = [l, A , . . . , A^" ] is an 299

300

Appendix E. The companion matrix

eigenvector of C ^ corresponding to the eigenvalue A. It is therefore simple in principle to construct a matrix having the form required by equation (4.49) and having the prescribed eigenvalues X^ merely by putting Pn (A) = (A — Ai) (A — A2)... (A — A^).