An Upper Bound for the Largest Eigenvalue of a Positive Semidefinite Block Banded Matrix

The new upper bound

$$ {\uplambda}_{\mathrm{max}}(A)\le \sum \limits_{k=1}^{p+1}i\equiv {k}_{\left(\operatorname{mod}p+1\right)}^{\mathrm{max}}{\uplambda}_{\mathrm{max}}\left({A}_{ii}\right) $$

for the largest eigenvalue of a Hermitian positive semidefinite block banded matrix A = (A_ij ) of block semibandwidth p is suggested. In the special case where the diagonal blocks of A are identity matrices, the latter bound reduces to the bound

$$ {\uplambda}_{\mathrm{max}}(A)\le p+1, $$

depending on p only, which improves the bounds established for such matrices earlier and extends the bound

$$ {\uplambda}_{\mathrm{max}}(A)\le 2, $$

old known for p = 1, i.e., for block tridiagonal matrices, to the general case p ≥ 1.