MHB Sava's question via email about symmetric matrices

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The discussion focuses on proving that the product of a matrix and its transpose, C^T C, is symmetric. A matrix is symmetric if it equals its own transpose, so the goal is to show that (C^T C)^T = C^T C. By applying the property of transposes, (MN)^T = N^T M^T, the proof demonstrates that (C^T C)^T simplifies to C^T C. This confirms that C^T C is indeed a symmetric matrix for any matrix C. The conclusion is that C^T C is symmetric, as required.
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Use the result $\displaystyle \begin{align*} \left( M\,N \right) ^T = N^T\,M^T \end{align*}$ to prove that for any matrix $\displaystyle \begin{align*} C \end{align*}$, $\displaystyle \begin{align*} C^T\,C \end{align*}$ is a symmetric matrix.

A matrix is symmetric if it is equal to its own transpose, so to show $\displaystyle \begin{align*} C^T\,C \end{align*}$ is symmetric, we need to prove that $\displaystyle \begin{align*} \left( C^T\,C \right) ^T = C^T\,C \end{align*}$.

$\displaystyle \begin{align*} \left( C^T\,C \right) ^T &= C^T\,\left( C^T \right) ^T \textrm{ as } \left( M\,N \right) ^T = N^T\,M^T \\ &= C^T\,C \end{align*}$

Since for any matrix $\displaystyle \begin{align*} C \end{align*}$, $\displaystyle \begin{align*} \left( C^T\,C \right) ^T = C^T\,C \end{align*}$, that means $\displaystyle \begin{align*} C^T\,C \end{align*}$ is a symmetric matrix.
 
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