MHB Sava's question via email about symmetric matrices

[Prove It]

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.

 

[Mark44]

QED...