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.