why the Eigen values of given matrix P are of modulus $1$?

Question

If P be a $2\times2$ complex matrix such that $P^*P=I$, where P* is the conjugate transpose of P. Then why the eigen values of P are of modulus $1$?

see here http://math.stackexchange.com/questions/1717713/show-that-the-eigenvalues-of-a-unitary-matrix-have-modulus-1/1717735#1717735 — Bérénice, Apr 09 '16 at 11:12

score 3 · Accepted Answer · answered Apr 09 '16 at 11:14

3

Suppose $Pv = \lambda v$.Then $P^{*}v = \overline{\lambda}v$. Hence $v = PP^{*}v = \overline{\lambda}Pv = |\lambda|^{2}v$, where $v$ is nonzero.

answered Apr 09 '16 at 11:14

score 1 · Answer 2 · answered Apr 09 '16 at 11:08

1

Hint. If $(\lambda,x)$ is an eigenpair of $P$, consider $x^\ast P^\ast Px$.

answered Apr 09 '16 at 11:08

user1551

2 Answers2