Question

TRUE OR FALSE

If$\mathbf{\left|}\mathbf{\right|}\mathit{A}\overrightarrow{\mathbf{x}}\mathbf{\left|}\mathbf{\right|}\mathbf{⩽}\mathbf{\left|}\mathbf{\right|}\overrightarrow{\mathbf{x}}\mathbf{\left|}\mathbf{\right|}$for $\overrightarrow{\mathbf{x}}$all in ${\mathit{R}}^{\mathbf{n}}$, then Amust represent the orthogonal projection onto a subspace Vof ${\mathit{R}}^{\mathbf{n}}$.

Short answer

The given statement is false.

Step-by-step solution

Determine the definition of an orthogonal matrix.

A square matrix with real numbers or elements is said to be an orthogonal matrix if its transpose is equal to its inverse matrix.

Or it can be said that, when the product of a square matrix and its transpose gives an identity matrix, then the square matrix is known as an orthogonal matrix.

Consider an example

For example,

Suppose A is a square matrix with real elements and of $n\times n$order and $A^T$ is the transpose of A. Then according to the definition, if, $A^T=A^{-1}$is satisfied, then,

$A·A^T=I$

For example, take $$\begin{gathered} A=\left[1/20 \\ 0-1/2\right] \end{gathered}$$

By observing that we get,

$$\begin{gathered} ||A\left[x \\ y\right]||=\frac{1}{2}||\left[x \\ y\right]|| \\ ⩽||\left[x \\ y\right]|| \end{gathered}$$

But A doesn’t represent an orthogonal projection onto a subspace V of $R^n$.

Hence, the given statement is false.