16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.4.3

The key insight is that $O(n, \mathbb{C})$ consists of matrices that preserve the standard inner product—that's literally what orthogonality means. When you apply an orthogonal matrix to a vector, the norm (sum of squared components) stays unchanged. Think of it geometrically: orthogonal transformations are rotations and reflections, which don't stretch or shrink vectors. You just need to translate "orthogonal matrix" into the algebraic condition $M^* M = I$ and apply it to the sum $\sum_j y_j^2$.

The key insight is that $O(n, \mathbb{C})$ consists of matrices that preserve the standard inner product—that's literally what orthogonality means. When you apply an orthogonal matrix to a vector, the norm (sum of squared components) stays unchanged. Think of it geometrically: orthogonal transformations are rotations and reflections, which don't stretch or shrink vectors. You just need to translate "orthogonal matrix" into the algebraic condition $M^* M = I$ and apply it to the sum $\sum_j y_j^2$.