Gram Matrix

Definition

${\mathbf{A}}^{†}\mathbf{A}$

Facts

A gram matrix is Hermitian Matrix. So Diagonalizable Matrix

The Eigenvalues of real-valued gram matrix ${\mathbf{A}}^{⊺}\mathbf{A}$ is non-negative real.

Proof $||\mathbf{Ax}|{|}^2={\mathbf{x}}^{⊺}{\mathbf{A}}^{⊺}\mathbf{Ax}={\mathbf{x}}^{⊺}\lambda \mathbf{x}=\lambda {\mathbf{x}}^{⊺}\mathbf{x}=\lambda ||\mathbf{x}|{|}^2\Rightarrow \lambda =\frac{||\mathbf{Ax}|{|}^2}{||\mathbf{x}|{|}^2}$ Since the matrix ${\mathbf{A}}^{⊺}\mathbf{A}$ is symmetric and the eigenvector $\mathbf{x}\ne 0$, $0\le \lambda \in \mathbb{R}$