# 1.1.5 Inner and outer products

The *inner product* is the same as the dot product

To multiply the following matrices, one of them must be transposed

$$
U = \begin{bmatrix}
u_1 \\
u_2 \\
u_3 \\
\end{bmatrix}
V = \begin{bmatrix} v_1 \\
v_2 \\
v_3 \\
\end{bmatrix} \\
U^T = \begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix}\\
U^T V = u_1 v_1 + u_2 v_2 + u_3 v_3
$$

If $U^T V = 0$ the matrix is sait to be *orthogonal*

The *norm* of a vector is defined as

$$
||U|| = \sqrt{U^T U} = \sqrt{u_1^2 + u_2^2 + u_3^2}
$$

A matrix is said to be *normalised* if $||U|| = 1$

If a matrix is both orthogonal and normalised, it is *orthonormal*

The *outer product* is the opposite of the inner product

$$
U V^T = \begin{bmatrix}
u_1v_1 & u_1v_2 & u_1v_3 \\
u_2v_1 & u_2v_2 & u_2v_3 \\
u_3v_1 & u_3v_2 & u_3v_3 \\
\end{bmatrix}
$$