The Orthogonal Inner Product

A linear transformation
$T$
on a vector space
$V$
is orthogonal if for all vectors
$\mathbf{v} \in V$
,
$\left| T \mathbf{v} \right| = \left| \mathbf{v} \right|$
.
All rotations and reflections are orthogonal, as are any sequence of rotations and reflections. Because the transformation
$T$
preserves the lengths or magnitudes of vectors, the associated matrix must have determinant 1 and the columns must have magnitude 1 and any two columns must be orthogonal and have dot product zero.
Example: Let a linear transformation have associated matrix
$\left( \begin{array}{cc} cos \theta & -sin \theta \\ sin \theta & cos \theta \end{array} \right)$
.
This matrix represents a rotation of
$\mathbb{R}^2$
anticlockwise by an angle
$\theta$
.
The columns are the vectors
$\mathbf{v}_1 = \begin{pmatrix}cos \theta\\sin \theta \end{pmatrix} , \: \mathbf{v}_2 = \begin{pmatrix}- sin \theta\\cos \theta \end{pmatrix}$
.
$\left| \mathbf{v}_1 \right| = cos^2 \theta + sin^2 \theta =1$

$\left| \mathbf{v}_2 \right| = (-sin \theta)^2 + cos^2 \theta =1$

and
$\mathbf{v}_1 \cdot \mathbf{v}_2 = cos \theta \times - sin \theta + sin \theta \times cos \theta =0$

We can also define an inner product on
$V$
by
$\langle \mathbf{v}_1 , \mathbf{v}_2 \rangle = \mathbf{v}_1^T M \mathbf{v}_2$
where
$M$
is the matrix associated with the orthogonal transformation
$T$
.
The inner product defined in this way has all the required properties, of being symmetric, positive definite and linear in both arguments.