Wikipedia:Reference desk/Archives/Mathematics/2017 July 5

= July 5 =

Eigenvalues of A^2
Hello,

It's straightforward to show that if λ is an eigenvalue of A then λ^2 is an eigenvalue of A^2. I'm trying to prove that opposite, that if λ^2 an eigenvalue of A^2 (over the complex numbers C) then λ or -λ is an eigenvalue of A. I would appreciate any direction. — Preceding unsigned comment added by 77.127.22.176 (talk) 15:32, 5 July 2017 (UTC)

$$\det\left[A^2 - \lambda^2 I\right] = \det\left[\left(A - \lambda I\right)\left(A + \lambda  I\right)\right] = \det\left[A - \lambda I\right] \det\left[A + \lambda I\right] $$ Count Iblis (talk) 20:55, 5 July 2017 (UTC)
 * Could you pleas elaborate? why it cannot be the case that (x-λ^2) appears in the characteristic polynomial of A-λI or A+λI? — Preceding unsigned comment added by 77.127.22.176 (talk) 06:08, 6 July 2017 (UTC)
 * never mind, I understand. Thanks! — Preceding unsigned comment added by 77.127.22.176 (talk) 06:42, 6 July 2017 (UTC)
 * By a similar argument you get a generalization: If p is a polynomial and μ is an eigenvalue of p(A), then every eigenvalue of A is a solution of p(λ)=μ.--RDBury (talk) 12:58, 7 July 2017 (UTC)
 * Shouldn't that be, instead of "every eigenvalue of A is", "there exists an eigenvalue of A which is"? -- Meni Rosenfeld (talk) 23:50, 8 July 2017 (UTC)
 * Yes. Good catch. --RDBury (talk) 10:04, 10 July 2017 (UTC)