On the Index of Block Upper Triangular Matrices
Rafael Bru, Joan‐Josep Climent, Michael Neumann · SIAM Journal on Matrix Analysis and Applications · 1995
Let M be an upper block triangular matrix with A and B singular diagonal blocks. It is known that $\max \{ \operatorname{index} ( A ), \operatorname{index} ( B ) \} \leq \operatorname{index} ( M ) \leq \operatorname{index} ( A ) + \operatorname{index} ( B )$. Recently, a necessary and sufficient condition has been given so that $\operatorname{index}( M ) = \operatorname{index}( A ) + \operatorname{index}( B )$. In this paper we find various characterizations for $\operatorname{index}( M )$ to take any specific values between $\max \{ \operatorname{index} ( A ), \operatorname{index} ( B ) \}$ and $\operatorname{index} ( A ) + \operatorname{index} ( B )$ which generalize previous results.