# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Formulas for calculating the extremum ranks and inertias of a four-term quadratic matrix-valued function and their applications. (English) Zbl 1252.15026

Let $A$ be an $m×m$ complex Hermitian matrix, $B$ be an $m×n$ complex matrix, $C$ be an $n×m$ complex Hermitian matrix, $D$ be an $n×p$ complex matrix, $X$ be a $p×m$ variable matrix and ${\left(·\right)}^{*}$ denotes the conjugate transpose of a complex matrix.

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 $DXA{X}^{*}{D}^{*}+DXB+{B}^{*}{X}^{*}{D}^{*}+C$. 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.

##### MSC:
 15A54 Matrices over function rings 15A24 Matrix equations and identities 15A63 Quadratic and bilinear forms, inner products 15B57 Hermitian, skew-Hermitian, and related matrices 90C20 Quadratic programming 90C22 Semidefinite programming 15B48 Positive matrices and their generalizations; cones of matrices 15A18 Eigenvalues, singular values, and eigenvectors 65K05 Mathematical programming (numerical methods) 15A45 Miscellaneous inequalities involving matrices 15A03 Vector spaces, linear dependence, rank