What can we say about $A$ if $AA^T$ is singular?

Question

I have a matrix $A$ of dimensions 100x50. It turns out $AA^T$ is actually singular. I obtained the matrix $A$ as the load matrix in a Factor Analysis output. This is the first time I have encountered such a problem in Factor Analysis.

What special properties of A leads to this singularity?

Answer 1

If a matrix $A$ has $100$ rows and $50$ columns, then $AA^T$ has $100$ rows and $100$ columns but its rank is no more than $50.$ Hence it will be singular regardless of what matrix $A$ is.

On the other hand, if $A$ has fewer rows than columns, then $AA^T$ will be singular if and only if the rows of $A$ are linearly dependent.

If you say somewhat more about your factor analysis problem, I might say something about what this means for that.

Answer 2

Since $A^T$ has dimensions $50\times 100$ we have that for some $x\neq 0$

$$A^Tx=0$$

and thus

$$AA^Tx=0$$

therefore $AA^T$ is always singular.