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Let it be two $m \times n$ matrices: $A$ and $B$, where $m,n \geq2$.
Rows of these matrices are linearly independent.

So, which matrix is guaranteed invertible: $AA^T$, $B^TB$, $AB^T$, $A^TB$?

I came to the idea that $A^TB$ and $B^TB$ can't be invertable in case $n>m$.
But need help with choose between $AB^T$ and $AA^T$

Georgii
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1 Answers1

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As you have rightly said, $B^TB$ and $A^TB$ need not be invertible. Similarly, $AB^T$ can also be noninvertible. For example, when rows of A and B are orthogonal to each other: \begin{equation} A = \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ \end{bmatrix} \hspace{0.5 cm}B= \begin{bmatrix} 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix}. \end{equation}

However, Rank$\{A\}$=Rank$\{AA^T\}=m.$ Hence, $AA^T$ is invertible. Refer this answer for the proof: Prove rank $A^TA$ = rank $A$ for any $A_{m \times n}$

Explorer
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