Let Q be an orthogonal matrix (a) Prove that if is an eigenvalue, then so is 1 (b) Prove that all its eigenvalues are complex numbers of modulus 1 In particular, the only possible real eigenvalues of an orthogonal matrix are 1 (c) Suppose v x iy is a complex eigenvector corresponding to a non real ...

a If Qv v then Q T v Q 1 v 1 v and so 1 is an eigenvalue of ...

Let Q be an orthogonal matrix. (a) Prove that if is an eigenvalue, then so is

Question:

Let Q be an orthogonal matrix.
(a) Prove that if λ is an eigenvalue, then so is 1/λ.
(b) Prove that all its eigenvalues are complex numbers of modulus |λ| = 1. In particular, the only possible real eigenvalues of an orthogonal matrix are ±1.
(c) Suppose v = x + iy is a complex eigenvector corresponding to a non-real eigenvalue. Prove that its real and imaginary parts are orthogonal vectors having the same Euclidean norm.