Matrices Multiplication And Inner Product

Question

On Wikipedia it says that "The Euclidean inner product and outer product are the simplest special cases of the matrix product"

But, can we look at matrices multiplication as an "expansion" of inner product? meaning that matrices multiplication is in particular an inner product on each row and column?

Answer 1

Not an answer, but might be interesting.

Let $A$ and $B$ be real matrices, then $(AB)_{ij}=\sum_{k}a_{ik}b_{kj}=\left\langle R_i(A), C_j(B)^T \right\rangle$ where $R_i(A)$ denotes the $i$-th row of $A$ and $C_j(B)$ denotes the $j$-th column of $B$ and $\left\langle -,-\right\rangle$ denotes the standard inner product on $\mathbb{R}^n$. (So indeed, a matrix product is nothing but a bunch of inner-products).

Now suppose $A$ is a real $n\times n$-matrix and $AA^T=I$. Then $(AA^T)_{ij}=\delta_{ij}$. Now notice that $C_i(A^T)^T=R_i(A)$ by definition of $A^T$. Thus $(AA^T)_{ij}=\left\langle R_i(A), R_j(A) \right\rangle=\delta_{ij}$. It follows that the rows of $A$ form on orthonormal basis of $\mathbb{R}^n$. This also explains why a square matrix satisfying $AA^T=I$ is called orthogonal.

This shows that this way of thinking about matrix multiplication can be interesting. (For example: try to find the analogues for complex matrices).