Positive-definite matrix submatrix

Question

Let $A \in \mathbb R^{n\times n}\;$be a positive-defnite matrix then $A^k \in \mathbb R^{k\times k} \quad A^k_{i,j}=A_{i,j} \quad 1\le i,j \le k\quad $ is also a positive defnite matrix. Let $x_k,\in \mathbb R_k:=\{x\in\mathbb R^n:x=(x_1,x_2,...x_k,0,...0)\}$

If one looks at $\; x_k^tAx_k>0\;$ which I think is $x^t\hat Ax>0\;$ $\hat A \in \mathbb R^{n \times n}\;$is just $A^k\;$ but you fill everything else with $0$. Is this true and if so does this imply that $A^k\;$is positive-definite ?