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In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

One way to express this is. Where QT is the transpose of Q and I is the identity matrix.

For example,

$Q^{T}Q=\left(\begin{array}{c}{{\stackrel{鈫�}{q}}_1}^T\\ {{\stackrel{鈫�}{q}}_2}^T\\ {{\stackrel{鈫�}{q}}_3}^T\\ .\\ .\\ {{\stackrel{鈫�}{q}}_n}^T\end{array}\right)({\stackrel{鈫�}{q}}_1,{\stackrel{鈫�}{q}}_2,{\stackrel{鈫�}{q}}_3,.......,{\stackrel{鈫�}{q}}_n)$

This subspace can be found using the Gram-Schmidt process.

Consider a basis of a subspace Vof ${\mathit{R}}^{\mathbf{n}}$for$\mathit{j}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}\mathit{m}$we resolve the vector ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{j}}$into its components parallel and perpendicular to the span of the preceding vectors ${\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{j}\mathbf{-}\mathbf{1}}$,

Then,

${\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{1}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\left|}\mathbf{\right|}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{\left|}\mathbf{\right|}}{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{1}}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{2}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\left|}\mathbf{\right|}{{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}}^{\mathbf{鈯�}}\mathbf{\left|}\mathbf{\right|}}{{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{2}}}^{\mathbf{鈯�}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{j}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\left|}\mathbf{\right|}{{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{j}}}^{\mathbf{鈯�}}\mathbf{\left|}\mathbf{\right|}}{{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{j}}}^{\mathbf{鈯�}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\stackrel{\mathbf{鈫�}}{\mathbf{u}}}_{\mathbf{m}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\left|}\mathbf{\right|}{{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{m}}}^{\mathbf{鈯�}}\mathbf{\left|}\mathbf{\right|}}{{\stackrel{\mathbf{鈫�}}{\mathbf{v}}}_{\mathbf{m}}}^{\mathbf{鈯�}}$

Thus, by using Gram-Schmidt process the nonzero subspace of has an orthonormal basis.

Hence, the statement is true.