Question: Let Q be an orthogonal matrix. (a) Prove that if is an eigenvalue, then so is 1/. (b) Prove that all its eigenvalues are

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.

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