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Let $A$ be an $m \times n$ matrix, and suppose $\vec{v}$ and $\vec{w}$ are orthogonal eigenvectors of $A^{T} A$. Show that $A \vec{v}$ and $A \vec{w}$ are orthogonal.
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Since $\vec{v}$ and $\vec{w}$ are orthogonal eigenvectors of $A^T A$, we have $A^T A \vec{v}=\lambda_v \vec{v}$ and $A^T A \vec{w}=\lambda_w \vec{w}$, where $\lambda_v$ and $\lambda_w$ are the corresponding eigenvalues.
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