| Title:
|
G-matrices, $J$-orthogonal matrices, and their sign patterns (English) |
| Author:
|
Hall, Frank J. |
| Author:
|
Rozložník, Miroslav |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
3 |
| Year:
|
2016 |
| Pages:
|
653-670 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $A^{\rm -T}= D_1 AD_2$, where $A^{\rm -T}$ denotes the transpose of the inverse of $A$. Denote by $J = {\rm diag}(\pm 1)$ a diagonal (signature) matrix, each of whose diagonal entries is $+1$ or $-1$. A nonsingular real matrix $Q$ is called $J$-orthogonal if $Q^{\rm T}J Q=\nobreak J$. 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. (English) |
| Keyword:
|
G-matrix |
| Keyword:
|
$J$-orthogonal matrix |
| Keyword:
|
Cauchy matrix |
| Keyword:
|
sign pattern matrix |
| MSC:
|
15A15 |
| MSC:
|
15A23 |
| MSC:
|
15A80 |
| idZBL:
|
Zbl 06644025 |
| idMR:
|
MR3556859 |
| DOI:
|
10.1007/s10587-016-0284-8 |
| . |
| Date available:
|
2016-10-01T15:15:07Z |
| Last updated:
|
2023-10-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145863 |
| . |
| Reference:
|
[1] Beasley, L. B., Scully, D. J.: Linear operators which preserve combinatorial orthogonality.Linear Algebra Appl. 201 (1994), 171-180. Zbl 0802.05016, MR 1274886 |
| Reference:
|
[2] Brualdi, R. A., Ryser, H. J.: Combinatorial Matrix Theory.Encyclopedia of Mathematics and Its Applications 39 Cambridge University Press, Cambridge (1991). Zbl 0746.05002, MR 1130611 |
| Reference:
|
[3] Brualdi, R. A., Shader, B. L.: Matrices of Sign-Solvable Linear Systems.Cambridge Tracts in Mathematics 116 Cambridge University Press, Cambridge (1995). Zbl 0833.15002, MR 1358133 |
| Reference:
|
[4] Della-Dora, J.: Numerical linear algorithms and group theory.Linear Algebra Appl. 10 (1975), 267-283. Zbl 0308.65025, MR 0383721, 10.1016/0024-3795(75)90074-9 |
| Reference:
|
[5] Elsner, L.: On some algebraic problems in connection with general eigenvalue algorithms.Linear Algebra Appl. 26 (1979), 123-138. Zbl 0412.15019, MR 0535682, 10.1016/0024-3795(79)90175-7 |
| Reference:
|
[6] Eschenbach, C. A., Hall, F. J., Harrell, D. L., Li, Z.: When does the inverse have the same sign pattern as the transpose?.Czech. Math. J. 49 (1999), 255-275. Zbl 0954.15013, MR 1692477, 10.1023/A:1022496101277 |
| Reference:
|
[7] Fiedler, M.: Notes on Hilbert and Cauchy matrices.Linear Algebra Appl. 432 (2010), 351-356. Zbl 1209.15029, MR 2566483 |
| Reference:
|
[8] Fiedler, M.: Theory of Graphs and Its Applications.Proc. Symp., Smolenice, 1963 Publishing House of the Czechoslovak Academy of Sciences Praha (1964). MR 0172259 |
| Reference:
|
[9] Fiedler, M., Hall, F. J.: G-matrices.Linear Algebra Appl. 436 (2012), 731-741. Zbl 1237.15024, MR 2854903 |
| Reference:
|
[10] Fiedler, M., Markham, T. L.: More on G-matrices.Linear Algebra Appl. 438 (2013), 231-241. Zbl 1255.15035, MR 2993378 |
| Reference:
|
[11] Hall, F. J., Li, Z.: Sign pattern matrices.Handbook of Linear Algebra Chapman and Hall/CRC Press Boca Raton (2013). |
| Reference:
|
[12] Higham, N. J.: $J$-orthogonal matrices: properties and generation.SIAM Rev. 45 (2003), 504-519. Zbl 1034.65026, MR 2046506, 10.1137/S0036144502414930 |
| Reference:
|
[13] Rozložník, M., Okulicka-Dłużewska, F., Smoktunowicz, A.: Cholesky-like factorization of symmetric indefinite matrices and orthogonalization with respect to bilinear forms.SIAM J. Matrix Anal. Appl. 36 (2015), 727-751. Zbl 1317.65105, MR 3355770, 10.1137/130947003 |
| Reference:
|
[14] Waters, C.: Sign patterns that allow orthogonality.Linear Algebra Appl. 235 (1996), 1-13. MR 1374247, 10.1016/0024-3795(94)00098-0 |
| . |
