Properties of the correlation of a rank deficient matrix. Suppose $A$ is of dimension $m\times n$, and it is rank deficient. Then $A^TA$ is also rank deficient. Are their any other immediate properties of $A^TA$ that are known? (The reason I am asking is that I read in an article that high values outside of the diagonal of a correlation matrix imply that the data matrix is low rank and am not sure of this).

> **Fact:** For any matrix $A$, $A$ will have the same nullspace as $A^TA$.

Note that if $m \geq n$, then $A$ is rank-deficient if and only if $A$ has a non-trivial nullspace. $A^TA$ is a square matrix, so we similarly conclude that if $A^TA$ has a non-trivial nullspace, it is rank-deficient. So in this case, $A$ is rank deficient if and only if $A^TA$ is rank-dificient.

On the other hand, if $m < n$, $A$ will automatically have a non-trivial nullspace, so $A^TA$ will be rank deficient whether or not $A$ happens to be as well.

In either case, the desired implication holds.

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A similarly useful fact is the following: for any matrix $A$, $A$ will have the same column space as $AA^T$.