Abstract
A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D 1 and D 2 such that A −T = D 1 AD 2, where A −T denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or −1. A nonsingular real matrix Q is called J-orthogonal if Q T JQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided.
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References
L. B. Beasley, D. J. Scully: Linear operators which preserve combinatorial orthogonality. Linear Algebra Appl. 201 (1994), 171–180.
R. A. Brualdi, H. J. Ryser: Combinatorial Matrix Theory. Encyclopedia of Mathematics and Its Applications 39, Cambridge University Press, Cambridge, 1991.
R. A. Brualdi, B. L. Shader: Matrices of Sign-Solvable Linear Systems. Cambridge Tracts in Mathematics 116, Cambridge University Press, Cambridge, 1995.
J. Della-Dora: Numerical linear algorithms and group theory. Linear Algebra Appl. 10 (1975), 267–283.
L. Elsner: On some algebraic problems in connection with general eigenvalue algorithms. Linear Algebra Appl. 26 (1979), 123–138.
C. A. Eschenbach, F. J. Hall, D. L. Harrell, Z. Li: When does the inverse have the same sign pattern as the transpose? Czech. Math. J. 49 (1999), 255–275.
M. Fiedler: Notes on Hilbert and Cauchy matrices. Linear Algebra Appl. 432 (2010), 351–356.
M. Fiedler, ed.: Theory of Graphs and Its Applications. Proc. Symp., Smolenice, 1963. Publishing House of the Czechoslovak Academy of Sciences, Praha, 1964.
M. Fiedler, F. J. Hall: G-matrices. Linear Algebra Appl. 436 (2012), 731–741.
M. Fiedler, T. L. Markham: More on G-matrices. Linear Algebra Appl. 438 (2013), 231–241.
F. J. Hall, Z. Li: Sign pattern matrices. Handbook of Linear Algebra. Chapman and Hall/CRC Press, Boca Raton, 2013.
N. J. Higham: J-orthogonal matrices: properties and generation. SIAM Rev. 45 (2003), 504–519.
M. Rozložník, F. Okulicka-Dlużewska, A. Smoktunowicz: Cholesky-like factorization of symmetric indefinite matrices and orthogonalization with respect to bilinear forms. SIAM J. Matrix Anal. Appl. 36 (2015), 727–751.
C. Waters: Sign patterns that allow orthogonality. Linear Algebra Appl. 235 (1996), 1–13.
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This paper is dedicated to the memory of Professor Miroslav Fiedler; it was an honor to work with him. He was an exceptionally kind person, a wonderful friend, a tremendous inspiration, and a great mathematician.
This research is supported by the Grant Agency of the Czech Republic under the project 108/11/0853.
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Hall, F.J., Rozložník, M. G-matrices, J-orthogonal matrices, and their sign patterns. Czech Math J 66, 653–670 (2016). https://doi.org/10.1007/s10587-016-0284-8
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DOI: https://doi.org/10.1007/s10587-016-0284-8