Question
Prove: If $A$ is an $m \times n$ matrix, then $A^{T} A$ and $A A^{T}$ have the same rank.
Step 1
Then $A^{T}$ is an $n \times m$ matrix. Show more…
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Prove: If $\mathbf{u}$ and $\mathbf{v}$ are $n \times 1$ matrices and $A$ is an $n \times n$ matrix, then $$ \left(\mathbf{v}^{T} A^{T} A \mathbf{u}\right)^{2} \leq\left(\mathbf{u}^{T} A^{T} A \mathbf{u}\right)\left(\mathbf{v}^{T} A^{T} A \mathbf{v}\right) $$
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