\[\mathbf{v}_1, \: \mathbf{v}_2\]
are vectors in a vector space \[V\]
and \[T\]
is a linear transformation sending elements of the space onto other elements, with associated matrix \[M\]
, then \[\mathbf{v}^T_1 M \mathbf{v}_2 \]
is the inner product of \[v_1\]
with \[v_2\]
. \[T\]
may be any matrix, including the zero matrix.The inner product is symmetric:
\[\langle \mathbf{v}_1, \mathbf{v}_2 \rangle =\langle \mathbf{v}_2, \mathbf{v}_1 \rangle\]
The inner product is positive definite:
\[\langle \mathbf{v}, \mathbf{v} \rangle \geq 0\]
and \[\langle \mathbf{v}, \mathbf{v} \rangle = 0 \leftrightarrow \mathbf{v}= \mathbf{0}\]
The inner product is linear in both arguments:
\[\langle \alpha \mathbf{v}_1, \mathbf{v}_2 \rangle =\langle \mathbf{v}_1, \alpha \mathbf{v}_2 \rangle = \alpha \langle \mathbf{v}_1, \mathbf{v}_2 \rangle\]