Originally Posted by

**kalagota** $\displaystyle \( \implies \)$

suppose $\displaystyle A_{nxn}$ is a Gramian Matrix.

Then there exists a matrix $\displaystyle B_{nxn}$ such that $\displaystyle A = B^T B$

Let B be the matrix such that $\displaystyle A = B^T B$

$\displaystyle A^T = (B^T B)^T = B^T B = A$ *

hence, A is symmetric. (to show that all of its eigenvalues are non-negative, take an arbitrary symmetric matrix $\displaystyle A = B^T B$ and show that its eigenvalues are non-negative.)

* have you proven that $\displaystyle (AB)^T = B^T A^T$?