Let P denote an nxn matrix...(orthogonal question + converse)?

tip: working with ||Px||^2=||x||^2 is usually easier.

a)

suppose P is orthogonal.

then, (recall ||x|| = (x^t)x, where x^t means the transpose of x, and that (P^t)P = identity)

||Px||^2 = Px∙Px = [(Px)^t]Px = (x^t)(P^t)Px = (x^t)x = ||x||^2

b)

denote the ith row of P by p_i (and p_i^t its transpose)

suppose ||Px||^2=||x||^2 for every column x in Rⁿ.

first, set x = e_i, for i = 1, .. n,

where e_i is the matrix with '1' in the ith entry and zero otherwise,

thus, for i = 1, .., n,

1 = ||e_i||^2 = ||Pe_i||^2 = (p_i^t)p_i

now set x = e_i + e_j for i ≠ j,

LHS = ||P(e_i + e_j)||^2 = ... = 2 + 2(p_i^t)p_j

RHS = ||e_i + e_j||^2 = .. = 2

so (p_i^t)p_j = 0 for every i ≠ j.

thus P is orthogonal.

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