3.3 QR factorization

Orthogonal and ONC matrices

• orthogonal : \(\mathbf{u}^T\mathbf{v} = 0\)

• orthonormal : orthogonal + \(\mathbf{u}^T\mathbf{u} = 1\)

• ONC : A matrix who’s columns are an orthonormal collection.

Properties of \(n\times k\) matrix:

• \(\mathbf{Q}^T \mathbf{Q}= I\) (\(k\times k\) identity)

• \(||\mathbf{Q}\mathbf{x}||_2 = ||\mathbf{x}||_2\)

• \(||\mathbf{Q}||_2 = 1\)

• orthogonal matrix: A square ONC matrix

Suppose \(\mathbf{Q}\) is an \(n\times n\) real orthogonal matrix. Then: 1. \(\mathbf{Q}^T = \mathbf{Q}^{-1}\). 2. \(\mathbf{Q}^T\) is also an orthogonal matrix. 3. \(\kappa(\mathbf{Q})=1\) in the 2-norm. 4. For any other \(n\times n\) matrix \(\mathbf{A}\), \(\| \mathbf{A}\mathbf{Q} \|_2=\| \mathbf{A} \|_2\). 5. If \(\mathbf{U}\) is another \(n\times n\) orthogonal matrix, then \(\mathbf{Q}\mathbf{U}\) is also orthogonal.