Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1252.15026
Tian, Yongge
Formulas for calculating the extremum ranks and inertias of a four-term quadratic matrix-valued function and their applications. (English)
[J] Linear Algebra Appl. 437, No. 3, 835-859 (2012). ISSN 0024-3795

Let $A$ be an $m\times m$ complex Hermitian matrix, $B$ be an $m\times n$ complex matrix, $C$ be an $n\times m$ complex Hermitian matrix, $D$ be an $n\times p$ complex matrix, $X$ be a $p\times m$ variable matrix and $(.)^{*}$ denotes the conjugate transpose of a complex matrix. \par In this paper, the author presents a useful algebraic linearization method, which can convert the calculations of ranks and inertias of quadratic Hermitian matrix-valued functions (QHMF) into those of ranks and inertias of certain linear matrix-valued functions, then the author establishes a group of explicit formulas in closed form for calculating the global maximum and minimum ranks and inertias of this matrix-valued function with respect to the variable matrix $X$. As applications of these rank and inertia formulas, the author characterizes a variety of solvability conditions for some quadratic matrix equations and inequalities generated from $DXAX^{*}D^{*}+DXB+B^{*}X^{*}D^{*}+C$. In particular, the author gives analytical solutions to the two well-known classic optimization problems on the QHMF in the Löwner partial ordering. The results obtained and the techniques adopted for solving the matrix rank and inertia optimization problem enable us to make many new extensions of some classic results on quadratic forms, quadratic matrix equations and quadratic matrix inequalities.
[Süleyman Güler (Aydin)]

MSC 2000:: *15A54 Matrices over function rings
15A24 Matrix equations
15A63 Bilinear forms, etc.
15B57
90C20 Quadratic programming
90C22 Semidefinite programming
15B48
15A18 Eigenvalues of matrices, etc.
65K05 Mathematical programming (numerical methods)
15A45 Miscellaneous inequalities involving matrices
15A03 Vector spaces

Keywords: quadratic matrix-valued function; generalized algebraic Riccati equation; matrix inequality; rank; inertia; Löwner partial ordering; optimization; linearization method; complex Hermitian matrix; quadratic matrix equations; quadratic forms

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]