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P-matrix
In mathematics, a -matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P_0-matrices, which are the closure of the class of -matrices, with every principal minor \geq 0. Spectra of -matrices By a theorem of Kellogg, the eigenvalues of - and P_0- matrices are bounded away from a wedge about the negative real axis as follows: :If \ are the eigenvalues of an -dimensional -matrix, where n>1, then ::, \arg(u_i), < \pi - \frac,\ i = 1,...,n :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[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Z-matrix (mathematics)
In mathematics, the class of ''Z''-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form: :Z=(z_);\quad z_\leq 0, \quad i\neq j. Note that this definition coincides precisely with that of a negated Metzler matrix or quasipositive matrix, thus the term ''quasinegative'' matrix appears from time to time in the literature, though this is rare and usually only in contexts where references to quasipositive matrices are made. The Jacobian of a competitive dynamical system is a ''Z''-matrix by definition. Likewise, if the Jacobian of a cooperative dynamical system is ''J'', then (−''J'') is a ''Z''-matrix. Related classes are ''L''-matrices, ''M''-matrices, ''P''-matrices, ''Hurwitz'' matrices and ''Metzler'' matrices. ''L''-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a ''Z''-matrix is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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(''M'',''d'') have a unique solution. * Any P-matrix is a Q-matrix. Conversely, if a matrix is a Z-matrix and a Q-matrix, then it is also a P-matrix. See also *P-matrix In mathematics, a -matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P_0-matrices, which are the closure of the class of -matrices, with every principal minor \geq 0. Spectra of -matric ... * Z-matrix References * * * * Matrix theory Matrices (mathematics) {{matrix-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M-matrix
In mathematics, especially linear algebra, an ''M''-matrix is a matrix whose off-diagonal entries are less than or equal to zero (i.e., it is a ''Z''-matrix) and whose eigenvalues have nonnegative real parts. The set of non-singular ''M''-matrices are a subset of the class of ''P''-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices). The name ''M''-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive.. Characterizations An M-matrix is commonly defined as follows: Definition: Let be a real Z-matrix. That is, where for all . Then matrix ''A'' is also an ''M-matrix'' if it can be expressed in the form , where with , for all , where is at least as large as the maximum of the moduli of the eigenvalues of , and is an identity matrix. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Formulation Given a real matrix ''M'' and vector ''q'', the linear complementarity problem LCP(''q'', ''M'') seeks vectors ''z'' and ''w'' which satisfy the following constraints: * w, z \geqslant 0, (that is, each component of these two vectors is non-negative) * z^Tw = 0 or equivalently \sum\nolimits_i w_i z_i = 0. This is the complementarity condition, since it implies that, for all i, at most one of w_i and z_i can be positive. * w = Mz + q A sufficient condition for existence and uniqueness of a solution to this problem is that ''M'' be symmetric positive-definite. If ''M'' is such that has a solution for every ''q'', then ''M'' is a Q-matrix. If ''M'' is such that have a unique solution for every ''q'', then ''M'' is a P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 eigenvalue of largest magnitude and that eigenvalue is real. The corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems ( subshifts of finite type); to economics ( Okishio's theorem, Hawkins–Simon condition); to demography ( Leslie population age distribution model); to social networks ( DeGroot learning process); to Internet search engines (PageRank); and even to ranking of American football teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau. Statement Let positive and non-negative respectively describe matrices with exclusively positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Routh–Hurwitz Matrix
In mathematics, the Routh–Hurwitz matrix, or more commonly just Hurwitz matrix, corresponding to a polynomial is a particular matrix whose nonzero entries are coefficients of the polynomial. Hurwitz matrix and the Hurwitz stability criterion Namely, given a real polynomial :p(z)=a_z^n+a_z^+\cdots+a_z+a_n the n\times n square matrix : H= \begin a_1 & a_3 & a_5 & \dots & \dots & \dots & 0 & 0 & 0 \\ a_0 & a_2 & a_4 & & & & \vdots & \vdots & \vdots \\ 0 & a_1 & a_3 & & & & \vdots & \vdots & \vdots \\ \vdots & a_0 & a_2 & \ddots & & & 0 & \vdots & \vdots \\ \vdots & 0 & a_1 & & \ddots & & a_n & \vdots & \vdots \\ \vdots & \vdots & a_0 & & & \ddots & a_ & 0 & \vdots \\ \vdots & \vdots & 0 & & & & a_ & a_n & \vdots \\ \vdots & \vdots & \vdots & & & & a_ & a_ & 0 \\ 0 & 0 & 0 & \dots & \dots & \dots & a_ & a_ & a_n \end. is called Hurwitz matrix corresponding to the polynomial p. It was established by Adolf Hurwitz in 1895 that a real polynomial with a_0 > 0 is stable (that is, all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 for either of these, the notation \R_ or \R^ for \left\ and \R_^ or \R^_ for \left\ has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians. In a complex plane, \R_ is identified with the positive real axis, and is usually drawn as a horizontal ray. This ray is used as reference in the polar form of a complex number. The real positive axis corresponds to complex numbers z = , z, \mathrm^, with argument \varphi = 0. Properties The set \R_ is closed under addition, multiplication, and division. It inherits a topology from the real line and, thus, has the structure of a multiplicative topological group or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. * 2000, Order of the Rising Sun The is a Japanese honors system, Japanese order, established in 1875 by Emperor Meiji. The Order was the first national decoration awarded by the Japanese government, created on 10 April 1875 by decree of the Council of State. The badge feat ..., 3rd class. Published works Books * * * * Journal articles * * * * * * * * * References External links {{DEFAULTSORT:Nikaido, Hukukane 1923 births 2001 deaths 20th-century Japanese economists General equilibrium theorists University of Tokyo alumni Academic staff of Tokyo University of Science Academic staff of Hitotsubashi University University of Minnesota faculty University of California, Berkeley faculty University of Southern California faculty Academic st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Gale
David Gale (December 13, 1921 – March 7, 2008) was an American mathematician and economist. He was a professor emeritus at the University of California, Berkeley, affiliated with the departments of mathematics, economics, and industrial engineering and operations research. He has contributed to the fields of mathematical economics, game theory, and convex analysis. Personal life Gale graduated with a Bachelor of Arts from Swarthmore College, obtained a M.A. from the University of Michigan in 1947, and earned his Ph.D. in Mathematics at Princeton University in 1949. He taught at Brown University from 1950 to 1965 and then joined the faculty at the University of California, Berkeley. Gale lived in Berkeley, California, and Paris, France with his partner Sandra Gilbert, feminist literary scholar and poet. He has three daughters and two grandsons. Contributions Gale's contributions to mathematical economics include an early proof of the existence of competitive equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Matrix And Determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian. They are named after Carl Gustav Jacob Jacobi. The Jacobian matrix is the natural generalization to vector valued functions of several variables of the derivative and the differential of a usual function. This generalization includes generalizations of the inverse function theorem and the implicit function theorem, where the non-nullity of the derivative is replaced by the non-nullity of the Jacobian determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix. The Jacobian determinant is fundamentally use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
