Math Is Fun Forum

Not an answer, but might be interesting.

Let A and B be real matrices, then (AB)ij=∑kaikbkj=⟨Ri(A),Cj(B)T⟩ where Ri(A) denotes the i-th row of A and Cj(B) denotes the j-th column of B and ⟨−,−⟩ denotes the standard inner product on Rn. (So indeed, a matrix product is nothing but a bunch of inner-products).

Now suppose A is a real n×n-matrix and AAT=I. Then (AAT)ij=δij. Now notice that Ci(AT)T=Ri(A) by definition of AT. Thus (AAT)ij=⟨Ri(A),Rj(A)⟩=δij. It follows that the rows of A form on orthonormal basis of Rn. This also explains why a square matrix satisfying AAT=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).