Consider $M$,  a positive definite matrix. Let $M^{(1)}$ be the diagonal matrix which agrees with $M$ on the diagonal ($M_{ii}=M^{(1)}_{ii}$). We have that $M^{(1)}$ is positive definite because it is diagonalizable and it has non-negative eigenvalues.

What about general bands? Let  $M^{(b)}$ be the restriction of $M$ on a band: $M_{ij}^{(b)}=M_{ij}$ when  $i$  and $j$ differ by no more than $b$ in absolute value and  $M^{(b)}_{ij}=0$ for otherwise. Is $M^{(b)}$ positive definite for all $b$?