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

The new upper boundλmax(A)≤∑k=1p+1i≡k(modp+1)maxλmax(Aii) for the largest eigenvalue of a Hermitian positive semidefinite block banded matrix A = (Aij) 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 boundλ max(A) ≤ p+ 1 , depending on p only, which improves the bounds established for such matrices earlier and extends the boundλ max(A) ≤ 2 , old known for p = 1, i.e., for block tridiagonal matrices, to the general case p ≥ 1. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.