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On nonsingularity of block two-by-two matrices. (English) Zbl 1283.15009
Summary: We derive necessary and sufficient conditions for guaranteeing the nonsingularity of a block two-by-two matrix by making use of the singular value decompositions and the Moore-Penrose pseudoinverses of the matrix blocks. These conditions are complete, and much weaker and simpler than those given by D. W. Decker and H. B. Keller [J. Math. Anal. Appl. 75, 417–430 (1980; Zbl 0452.47075)], and may be more easily examined than those given by Z.-Z. Bai [J. Comput. Appl. Math. 237, No. 1, 295–306 (2013; Zbl 1252.15022)] from the computational viewpoint. We also derive general formulas for the rank of the block two-by-two matrix by utilizing either the unitarily compressed or the orthogonally projected sub-matrices.

MSC:
15A09 Theory of matrix inversion and generalized inverses
65F20 Numerical solutions to overdetermined systems, pseudoinverses
15A03 Vector spaces, linear dependence, rank, lineability
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