The maximal and minimal ranks of A - BXC with applications

New York Journal of Mathematics

Volumn 9, Issue , 2003, Pages 345-362

  Tian, Yongge ^a   Cheng, Shizhen ^b  

^a QUEEN S UNIVERSITY   (Canada)

^b TIANJIN POLYTECHNIC UNIVERSITY   (China)

Author keywords

Block matrix; Generalized inverse; Linear matrix expression; Maximal rank; Minimal rank; Range; Rank equation; Schur complement; Shorted matrix

Indexed keywords

EID: 3042645371     PISSN: 10769803     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (25)

1. W.N. Anderson Jr., Shorted operators, SIAM J. Appl. Math. 20 (1971), 520-525, MR 0287970, Zbl 0217.05503.
2. T. Ando, Generalized Schur complements, Linear Algebra Appl. 27 (1979), 173-186, MR 0545731, Zbl 0412.15006.
3. J.K. Baksalary and R. Kala, Range invariance of certain matrix products, Linear and Multilinear Algebra 14 (1986), 89-96, MR 0712827, Zbl 0523.15006.
4. D. Carlson, What are Schur complements, anyway?, Linear Algebra Appl. 74 (1986), 257-275, MR 0822151, Zbl 0595.15006.
5. N. Cohen, C.R. Johnson, L. Rodman and H.J. Woerdeman, Ranks of completions of partial matrices, Oper. Theory: Adv. Appl. 40 (1989), 165-185, MR 1038313, Zbl 0676.15001.
6. C. Davis, Completing a matrix so as to minimize the rank, Oper. Theory: Adv. Appl. 29 (1988), 87-95, MR 0945004, Zbl 0646.15001.
7. M. Fiedler, Remarks on the Schur complement, Linear Algebra Appl. 39 (1981), 189-195, MR 0625249, Zbl 0465.15004.
8. H. Goller, Shorted operators and rank decomposition matrices, Linear Algebra Appl. 81 (1986), 207-236, MR 0852902, Zbl 0597.15002.
9. J. Groß, Comments on range invariance of matrix products, Linear and Multilinear Algebra 41 (1996), 157-160, MR 1430488.
10. C.R. Johnson, Matrix completion problems: a survey, In Matrix Theory and Applications, Proc. Sympos. Appl. Math. AMS 40 (1990), 171-197, MR 1059486, Zbl 0706.15024.
11. C.R. Johnson and G.T. Whitney, Minimum rank completions, Linear and Multilinear Algebra 28 (1991), 271-273, MR 1088424, Zbl 0775.15001.
12. G. Marsaglia and G.P.H. Styan, Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra 2 (1974), 269-292, MR 0384840, Zbl 0297.15003.
13. C.D. Meyer, Jr., Generalized inverses and ranks of block matrices SIAM J. Appl. Math. 25 (1973), 597-602, MR 0330183, Zbl 0269.15002.
14. S.K. Mitra, The minus partial order and the shorted matrix, Linear Algebra Appl. 83 (1986), 1-27, MR 0862729, Zbl 0605.15004.
15. S.K. Mitra and M.L. Puri, Shorted matrices-an extended concept and some applications, Linear Algebra Appl. 42 (1982), 57-79, MR 0656414, Zbl 0478.15012.
16. R. Penrose, A generalized inverse for matrices, Proc. Camb. Philos. Soc. 51 (1955), 406-413, MR 0069793, Zbl 0065.24603.
17. Y. Tian, The general solutions of the matrix equation AXB = CY D, Math. Theory Practice 1 (1988), 71-73, MR 0957843.
18. Y. Tian, Completing triangular block matrices with maximal and minimal ranks, Linear Algebra Appl. 321 (2000), 327-345, MR 1800003, Zbl 0984.15013.
19. Y. Tian, Solvability of two linear matrix equations, Linear and Multilinear Algebra 48 (2000), 123-147, MR 1813440, Zbl 0970.15005.
20. Y. Tian, The minimum rank of a 3 × 3 partial block matrix., Linear and Multilinear Algebra 50 (2002), 125-131, MR 1892834, Zbl 1006.15004.
21. Y. Tian, The minimal rank of the matrix expression A - BX - YC, Missouri J. Math. Sci. 14 (2002), 40-48, MR 1883608.
22. Y. Tian, Upper and lower bounds for ranks of matrix expressions using generalized inverses, Linear Algebra Appl. 355 (2002), 187-214, MR 1930145, Zbl 1016.15003.
23. Y. Tian, The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math. 25 (2002), 745-755, MR 1934671, Zbl 1007.15005.
24. Y. Tian, The minimal rank of a block matrix with generalized inverses, Problem 29-11, IMAGE-The Bulletin of the International Linear Algebra Society 29 (2002), 35.
25. H.J. Woerdeman, Minimal rank completions for block matrices, Linear Algebra Appl. 121 (1989), 105-122, MR 1011731, Zbl 0681.15002.