Difference between quadratic forms obtained from a p.s.d matrix and its partition

Question

Let $A$ be a $n\times n$ positive semi definite matrix, with real valued entries. Partition $A$ as $$ A = \begin{pmatrix} A_{1,1} & A_{1,2}\\ A_{2,1} & A_{2,2} \end{pmatrix}, $$ where, $A_{1,1}$ is a $m\times m$ matrix. Let $\mathbf{x}\in \mathbb{R}^n$. Consider the difference, $$ g(\mathbf{x}) \equiv \mathbf{x}^\prime \left[ A - \begin{pmatrix} A_{1,1} & \mathbf{0}\\ \mathbf{0} & \mathbf{0} \end{pmatrix}\right]\mathbf{x}, $$ where, $\mathbf{0}$ denotes a matrix with all entries equal to zero. I want to show, $g(\mathbf{x})\geq 0$, for all $\mathbf{x}\in\mathbb{R}^n$. Is this true? Or, is it true only for symmetric and positive (semi)-definite $A$?