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A note on the eigenvalues of saddle point matrices

Liesen, Jörg

Inst. Mathematik

Results of Benzi and Simoncini (Numer. Math. 103 (2006), pp.~173--196) on spectral properties of block $2\times 2$ matrices are generalized to the case of a symmetric positive semidefinite block at the (2,2) position. More precisely, a sufficient condition is derived when a (nonsymmetric) saddle point matrix of the form $[A\;\;B^T; -B\;C]$ with $A=A^T>0$, full rank $B$, and $C=C^T\geq 0$, is diagonalizable and has real and positive eigenvalues.