Linear Algebra Question. Invertibility Problem.

Question

$A$ is a $d*k$ full rank matrix, given d > k. Is the matrix $I-AA^T$ invertible?

Answer 1

Take the $\;2\times1\;$ full rank matrix $\;\binom 10\;$ , then

$$\begin{pmatrix}1&0\\0&1\end{pmatrix}-\binom10(1\;0)=\begin{pmatrix}1&0\\0&1\end{pmatrix}-\begin{pmatrix}1&0\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\0&1\end{pmatrix}$$

Answer 2

$A A^T$ is positive semidefinite (assuming this is over the reals). If $\lambda > 0$ is an eigenvalue of $A A^T$, replace $A$ by $\lambda^{-1/2} A$ to get a counterexample.