Rank constrained matrix best approximation problem with respect to (skew) Hermitian matrices

In this literature, we study a rank constrained matrix approximation problem in the Frobenius norm: minr(X)=kBXBAF2, where k is a nonnegative integer, A and X are (skew) Hermitian matrices. By using the singular value decomposition and the spectrum decomposition, we derive some conditions for the existence of (skew) Hermitian solutions, and establish general forms for the (skew) Hermitian solutions to this matrix approximation problem.

[1]  Musheng Wei,et al.  Minimum rank (skew) Hermitian solutions to the matrix approximation problem in the spectral norm , 2015, J. Comput. Appl. Math..

[2]  Ivan Markovsky,et al.  Software for weighted structured low-rank approximation , 2014, J. Comput. Appl. Math..

[3]  Sujit Kumar Mitra,et al.  Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations , 1976 .

[4]  G. Styan,et al.  Equalities and Inequalities for Ranks of Matrices , 1974 .

[5]  Hongxing Wang,et al.  The minimal rank of matrix expressions with respect to Hermitian matrix-revised , 2016, J. Frankl. Inst..

[6]  E. Oja,et al.  Projection filter, Wiener filter, and Karhunen-Loève subspaces in digital image restoration , 1986 .

[7]  Peter Lancaster,et al.  Linear matrix equations from an inverse problem of vibration theory , 1996 .

[9]  Hongxing Wang.  Rank constrained matrix best approximation problem , 2015, Appl. Math. Lett..

[10]  Musheng Wei,et al.  On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation AXA H =B , 2007 .

[11]  Jeffrey L. Krolik,et al.  A generalized Karhunen-Loeve basis for efficient estimation of tropospheric refractivity using radar clutter , 2004, IEEE Transactions on Signal Processing.

[12]  J. Demmel.  The smallest perturbation of a submatrix which lowers the rank and constrained total least squares problems , 1987 .

[13]  Ivan Markovsky,et al.  Low Rank Approximation - Algorithms, Implementation, Applications , 2018, Communications and Control Engineering.

[14]  Tingzhu Huang,et al.  Extremal ranks of matrix expression of A - BXC with respect to Hermitian matrix , 2010, Appl. Math. Comput..

[15]  G. Stewart,et al.  A generalization of the Eckart-Young-Mirsky matrix approximation theorem , 1987 .

[16]  Hongxing Wang.  The minimal rank of A-BX with respect to Hermitian matrix , 2014, Appl. Math. Comput..

[17]  Yukihiko Yamashita,et al.  Relative Karhunen-Loève transform , 1996, IEEE Trans. Signal Process..

[18]  Hongxing Wang.  On least squares solutions subject to a rank restriction , 2015 .

[19]  A. Hoffman,et al.  The variation of the spectrum of a normal matrix , 1953 .

[20]  Jiao-fen Li,et al.  On the Low-Rank Approximation Arising in the Generalized Karhunen-Loeve Transform , 2013 .

[21]  Ying Li,et al.  A Hermitian least squares solution of the matrix equation AXB=C subject to inequality restrictions , 2012, Comput. Math. Appl..

[22]  Anatoli Torokhti,et al.  Generalized Rank-Constrained Matrix Approximations , 2007, SIAM J. Matrix Anal. Appl..

[23]  MUSHENG WEI,et al.  Minimum Rank Solutions to the Matrix Approximation Problems in the Spectral Norm , 2012, SIAM J. Matrix Anal. Appl..

[24]  A. Rantzer,et al.  On a generalized matrix approximation problem in the spectral norm , 2012 .

[25]  Yoshio Takane,et al.  Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA∗=B , 2009 .

[26]  Yongge Tian,et al.  Hermitian-type generalized singular value decomposition with applications , 2013, Numer. Linear Algebra Appl..

[27]  Wanquan Liu,et al.  Generalized Karhunen-Loeve transform , 1998, IEEE Signal Process. Lett..

[28]  Yongge Tian.  A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$ , 2015 .