The Re-nonnegative definite solutions to the matrix equation A X B = C

Qing Wen Wang; Chang Lan Yang

Commentationes Mathematicae Universitatis Carolinae (1998)

  • Volume: 39, Issue: 1, page 7-13
  • ISSN: 0010-2628

Abstract

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An n × n complex matrix A is called Re-nonnegative definite (Re-nnd) if the real part of x * A x is nonnegative for every complex n -vector x . In this paper criteria for a partitioned matrix to be Re-nnd are given. A necessary and sufficient condition for the existence of and an expression for the Re-nnd solutions of the matrix equation A X B = C are presented.

How to cite

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Wang, Qing Wen, and Yang, Chang Lan. "The Re-nonnegative definite solutions to the matrix equation $AXB=C$." Commentationes Mathematicae Universitatis Carolinae 39.1 (1998): 7-13. <http://eudml.org/doc/248253>.

@article{Wang1998,
abstract = {An $n\times n$ complex matrix $A$ is called Re-nonnegative definite (Re-nnd) if the real part of $x^\{\ast \} Ax$ is nonnegative for every complex $n$-vector $x$. In this paper criteria for a partitioned matrix to be Re-nnd are given. A necessary and sufficient condition for the existence of and an expression for the Re-nnd solutions of the matrix equation $AXB=C$ are presented.},
author = {Wang, Qing Wen, Yang, Chang Lan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Re-nonnegative define matrix; matrix equation; generalized singular value decomposition; Re-nonnegative definite matrix; matrix equation; generalized singular value decomposition},
language = {eng},
number = {1},
pages = {7-13},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The Re-nonnegative definite solutions to the matrix equation $AXB=C$},
url = {http://eudml.org/doc/248253},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Wang, Qing Wen
AU - Yang, Chang Lan
TI - The Re-nonnegative definite solutions to the matrix equation $AXB=C$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 1
SP - 7
EP - 13
AB - An $n\times n$ complex matrix $A$ is called Re-nonnegative definite (Re-nnd) if the real part of $x^{\ast } Ax$ is nonnegative for every complex $n$-vector $x$. In this paper criteria for a partitioned matrix to be Re-nnd are given. A necessary and sufficient condition for the existence of and an expression for the Re-nnd solutions of the matrix equation $AXB=C$ are presented.
LA - eng
KW - Re-nonnegative define matrix; matrix equation; generalized singular value decomposition; Re-nonnegative definite matrix; matrix equation; generalized singular value decomposition
UR - http://eudml.org/doc/248253
ER -

References

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  1. Wu L., Cain B., The Re-nonnegative definite solutions to matrix inverse problem A X = B , Linear Algebra Appl. 236 (1996), 137-146. (1996) MR1375611
  2. Khatri C.G., Mitra S.K., Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. Math. 31.4 (1976), 579-585. (1976) Zbl0359.65033MR0417212
  3. Chu K.E., Singular value and general singular value decompositions and the solution of linear matrix equation, Linear Algebra Appl. 88/89 (1987), 83-98. (1987) MR0882442
  4. Porter A.D., Mousouris N., Ranked solutions of A X C = B and A X = B , Linear Algebra Appl. 24 (1979), 217-224. (1979) Zbl0411.15009MR0524838
  5. Dai H., On the symmetric solution of linear matrix equations, Linear Algebra Appl. 131 (1990), 1-7. (1990) MR1057060
  6. Wang Q.W., The metapositive definite self-conjugate solutions of the matrix equation A X B = C over a skew field, Chinese Quarterly J. Math. 3 (1995), 42-51. (1995) 
  7. Wang Q.W., The matrix equation A X B = C over an arbitrary skew field, Chinese Quarterly J. Math. 4 (1996), 1-5. (1996) 
  8. Wang Q.W., Skewpositive semidefinite solutions to the quaternion matrix equation A X B = C , Far East. J. Math. Sci., to appear. MR1432967
  9. Paige C.C., Saunders M.A., Towards a generalized singular value decomposition, SIAM J. Numer. Anal. 18 (1981), 398-405. (1981) Zbl0471.65018MR0615522
  10. Stewart G.W., Computing the CS-decomposition of a partitioned orthogonal matrix, Numer. Math. 40 (1982), 297-306. (1982) MR0695598

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