Matrix Algebra

Orthonormal Basis

Orthonormal basis. A set $\{\mathbf{v}_1,\ldots,\mathbf{v}_k\}$ of nonzero vectors is called an orthogonal set if $\langle\mathbf{v}_i,\mathbf{v}_j\rangle = 0$ whenever $i \neq j$ . The orthogonal set is linearly independent. Thus, it is considered a basis for the subspace spanned by the orthogonal set; in this sense, it is called an orthogonal basis. An orthogonal basis $\{\mathbf{u}_1,\ldots,\mathbf{u}_k\}$ is called an orthonormal basis if $\Vert\mathbf{u}_i\Vert = 1$ for all . An orthnormal basis $\{\mathbf{u}_1,\ldots,\mathbf{u}_n\}$ of vectors in $\mathbb{R}^n$ becomes a basis for $\mathbb{R}^n$ . Furthermore, the $n\times n$ matrix

$\displaystyle U = [\mathbf{u}_1,\ldots,\mathbf{u}_n]$

is called orthogonal, and satisfies

. In fact, any matrix satisfying

is orthogonal, and the columns of

form an orthonormal basis for $\mathbb{R}^n$ .

EXAMPLE 2. Let

$\displaystyle U = \left[\begin{array}{rrr} 1/2 & 1/2 & 1/\sqrt{2} \\ 1/2 & 1/2 & -1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} & 0 \end{array}\right]$

be a $3\times 3$ square matrix. Then show that

is an orthogonal matrix.

EXAMPLE 3. Let $\mathbf{v}_1 = \left[\begin{array}{c} 1 \\ -2 \\ 1 \end{array}\right]$ , $\mathbf{v}_2 = \left[\begin{array}{c} 0 \\ 1 \\ 2 \end{array}\right]$ , and $\mathbf{v}_3 = \left[\begin{array}{c} -5 \\ -2 \\ 1 \end{array}\right]$ be orthogonal vectors in $\mathbb{R}^3$ . Construct the orthonormal basis $\mathbf{u}_1$ , $\mathbf{u}_2$ , and $\mathbf{u}_3$ , and the corresponding orthogonal matrix .