Math 511: Linear Algebra

Orthogonal Matrices and Change of Basis

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1. Show that the matrix $P = \begin{bmatrix} 3 & -2 \\ 2 & -1 \end{bmatrix}$ is not orthogonal.¶

2. Show that for any real number $\theta$, the matrix¶

$$ P = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} $$

is orthogonal.

3. Prove that a matrix is orthogonal if and only if it is a square matrix whose columns are pairwise orthonormal.¶

4. Prove that the inverse of an orthogonal matrix is orthogonal.¶

5. Answer the following¶

(a) Is the sum of orthogonal matrices orthogonal?

(b) Is the product of two orthogonal matrices orthogonal?

6. Prove that if $P$ is an $n\times n$ orthogonal matrix, $\|P\mathbf{x}\| = \|\mathbf{x}\|$ for all vectors $\mathbf{x}$ in $\mathbb{R}^n$.¶

7. Verify the result of exercise 6¶

using the bases $B = \left\{ \begin{bmatrix}1\\0\end{bmatrix},\ \begin{bmatrix}0\\1\end{bmatrix} \right\}$ and $B' = \left\{ \begin{bmatrix} -\frac{2}{\sqrt{5}} \\ \frac{1}{\sqrt{5}} \end{bmatrix},\ \begin{bmatrix} \frac{1}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} \end{bmatrix} \right\}$.

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