Let Q be an n times n real orthogonal matrix1 Justify that f

Let Q be an n times n real orthogonal matrix.^1 Justify that for every x, y R^n, (Qx) middot (Qy) = x middot y. Justify that for every x, y R^n, (Q(x - y)) middot (Q(x - y)) = (x - y) middot (x - y). Suppose that Q has real eigenvalues. Find all possible real eigenvalues of Q. Suppose that Q has real eigenvalues. Is it true Q must have all possible eigenvalues obtained in iii)? Justify your answer. Is it possible that an orthogonal matrix does not have any real eigenvalues? Justify your answer.^2

Solution

3) if a is an eigen vector of the matrix Q, then, (1/a) is also an eigen vector of Q.

4) If Q has real eigenvalues and if a is an eigen value, then Q must have(1/a) as its eigen value because the matrix Q is orthogonal.