Question-5

\(Q\) is an orthogonal matrix. If \(||Qu|| = 4\), what is \(||u||\)?

Hint

Expand \(||Qu||^2\) and use the key property of orthogonal matrices.

Answer

\(4\)

Solution

Orthogonal matrices preserve the inner product: \[ \begin{aligned} ||Qu||^2 &= (Qu)^T(Qu)\\ &= u^TQ^TQu\\ &= u^TIu\\ &= u^Tu\\ &= ||u||^2 \end{aligned} \]