How is dot product defined for higher dimensions?

Question

I am specifically interested in the case of 2D (matrices). I am reading a research paper which says that I should find the dot product of matrix A and its transpose and subtract it from the identity matrix.

$AA^T - I$

From the matrix library I am using the dot product comes out to be a single number?

Can anyone explain why it would be a matrix?

Thanks.

Answer 1

It's sort of weird for the paper to call the operation "dot product" because that name is usually reserved for the dot product of two vectors. What the paper is talking about, I believe, is simply the standard matrix multiplication which, if the matrix is a single column, is the same as the dot product.

So yeah, to describe the formula

$$AA^T-I$$

I would say you simply take the product of $A$ and $A^T$ and subtract $I$.