In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a -matrix is a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
with every principal
minor
Minor may refer to:
* Minor (law), a person under the age of certain legal activities.
** A person who has not reached the age of majority
* Academic minor, a secondary field of study in undergraduate education
Music theory
*Minor chord
** Barb ...
is positive. A closely related class is that of
-matrices, which are the closure of the class of -matrices, with every principal minor
0.
Spectra of -matrices
By a theorem of Kellogg, the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s of - and
- matrices are bounded away from a wedge about the negative real axis as follows:
:If
are the eigenvalues of an -dimensional -matrix, where
, then
::
:If
,
,
are the eigenvalues of an -dimensional
-matrix, then
::
Remarks
The class of nonsingular
''M''-matrices is a subset of the class of -matrices. More precisely, all matrices that are both -matrices and
''Z''-matrices are nonsingular -matrices. The class of
sufficient matrices is another generalization of -matrices.
The
linear complementarity problem In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle and Dantzig in 1968.
...
has a unique solution for every vector if and only if is a -matrix. This implies that if is a -matrix, then is a
-matrix.
If the
Jacobian of a function is a -matrix, then the function is injective on any rectangular region of
.
A related class of interest, particularly with reference to stability, is that of
-matrices, sometimes also referred to as
-matrices. A matrix is a
-matrix if and only if
is a -matrix (similarly for
-matrices). Since
, the eigenvalues of these matrices are bounded away from the
positive real axis
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
.
See also
*
Hurwitz matrix In mathematics, a Hurwitz matrix, or Routh–Hurwitz matrix, in engineering stability matrix, is a structured real square matrix constructed with coefficients of a real polynomial.
Hurwitz matrix and the Hurwitz stability criterion
Namely, given a ...
*
Linear complementarity problem In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle and Dantzig in 1968.
...
*
M-matrix In mathematics, especially linear algebra, an ''M''-matrix is a ''Z''-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular ''M''-matrices are a subset of the class of ''P''-matrices, and also of the class of inverse-p ...
*
Q-matrix
In mathematics, a Q-matrix is a square matrix whose associated linear complementarity problem LCP(''M'',''q'') has a solution for every vector ''q''.
Properties
* ''M'' is a Q-matrix if there exists ''d'' > 0 such that LCP(''M'',0) and LCP('' ...
*
Z-matrix
*
Perron–Frobenius theorem
In matrix theory, the Perron–Frobenius theorem, proved by and , asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive component ...
Notes
References
*
*
David Gale
David (; , "beloved one") (traditional spelling), , ''Dāwūd''; grc-koi, Δαυΐδ, Dauíd; la, Davidus, David; gez , ዳዊት, ''Dawit''; xcl, Դաւիթ, ''Dawitʿ''; cu, Давíдъ, ''Davidŭ''; possibly meaning "beloved one". w ...
and
Hukukane Nikaido
was a Japanese economist.
Career
He received a B.S. in mathematics from the University of Tokyo and a D.Sc. in mathematics from the University of Tokyo in 1961.
honors
* 1962, Fellow, Econometric Society
The Econometric Society is an inter ...
, The Jacobian matrix and global univalence of mappings, ''Math. Ann.'' 159:81-93 (1965)
* Li Fang, On the Spectra of - and
-Matrices, ''Linear Algebra and its Applications'' 119:1-25 (1989)
* R. B. Kellogg, On complex eigenvalues of and {{mvar, P matrices, ''Numer. Math.'' 19:170-175 (1972)
Matrix theory
Matrices