Question

TRUE OR FALSE

If A is an $\mathit{n}\mathbf{\times }\mathit{n}$matrix such that$||A\overrightarrow{u}||\mathbf{=}\mathbf{1}$ for all unit vectors $\overrightarrow{\mathbf{u}}$, then A must be an orthogonal matrix.

Short answer

The given statement is true.

Step-by-step solution

Definition of orthogonal matrix

Matrix A is orthogonal if and only if$x\in R^n,||A\overrightarrow{x}||=||\overrightarrow{x}||$.

Explanation for the given statement

Consider the statement as follows:

“If A is an$n\times n$matrix such that $||A\overrightarrow{u}||=1$for all unit vector $\overrightarrow{u}$, then A must be an orthogonal matrix.

Let $\overrightarrow{u}$is a unit vector such that $||A\overrightarrow{u}||=1$.

Then for any $x\in R^n$, $\overrightarrow{u}=\frac{\overrightarrow{x}}{||\overrightarrow{x}||}$.

$$\begin{gathered} ||A\overrightarrow{u}||=1 \\ ||A\left(\frac{\overrightarrow{x}}{||\overrightarrow{x}||}\right)||=1 \\ \frac{1}{||\overrightarrow{x}||}||A\overrightarrow{x}||=1 \\ ||A\overrightarrow{x}||=||\overrightarrow{x}|| \end{gathered}$$

Hence, A is orthogonal matrix.

Thus, the given statement “If A is an $n\times n$matrix such that$||A\overrightarrow{u}||=1$ for all unit vector $\overrightarrow{u}$, then A must be an orthogonal matrix” is true.