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 less than $b$ in absolute value and $M^{(b)}_{ij}=0$ for otherwise. Is $M^{(b)}$ positive definite for all $b$?