Totally Positive
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Totally Positive
In mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix is a positive number. A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly, except that all the minors must be non-negative (positive or zero). Some authors use "totally positive" to include all totally non-negative matrices. Definition Let \mathbf = (A_)_ be an ''n'' × ''n'' matrix. Consider any p\in\ and any ''p'' × ''p'' submatrix of the form \mathbf = (A_)_ where: : 1\le i_1 < \ldots < i_p \le n,\qquad 1\le j_1 <\ldots < j_p \le n. Then A is a totally positive matrix if:
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Positive Matrix
In mathematics, a nonnegative matrix, written : \mathbf \geq 0, is a matrix in which all the elements are equal to or greater than zero, that is, : x_ \geq 0\qquad \forall . A positive matrix is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is a subset of all non-negative matrices. While such matrices are commonly found, the term is only occasionally used due to the possible confusion with positive-definite matrices, which are different. A matrix which is both non-negative and is positive semidefinite is called a doubly non-negative matrix. A rectangular non-negative matrix can be approximated by a decomposition with two other non-negative matrices via non-negative matrix factorization. Eigenvalues and eigenvectors of square positive matrices are described by the Perron–Frobenius theorem. Properties *The trace and every row and column sum/product of a nonnegative matrix is nonnegative. Inversion The inverse of any non-singul ...
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Spectrum (functional Analysis)
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number \lambda is said to be in the spectrum of a bounded linear operator T if T-\lambda I is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the right shift operator ''R'' on the Hilbert space ℓ2, :(x_1, x_2, \dots) \mapsto (0, x_1, x_2, \dots). This has no eigenvalues, since if ''Rx''=''λx'' then by expanding this expression we see ...
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Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ...
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Compound Matrix
In linear algebra, a branch of mathematics, a (multiplicative) compound matrix is a matrix whose entries are all minors, of a given size, of another matrix. Compound matrices are closely related to exterior algebras, and their computation appears in a wide array of problems, such as in the analysis of nonlinear time-varying dynamical systems and generalizations of positive systems, cooperative systems and contracting systems. Definition Let be an matrix with real or complex entries. If is a subset of size of and is a subset of size of , then the -submatrix of , written  , is the submatrix formed from by retaining only those rows indexed by and those columns indexed by . If , then is the - minor of . The ''r'' th compound matrix of is a matrix, denoted , is defined as follows. If , then is the unique matrix. Otherwise, has size \binom \!\times\! \binom. Its rows and columns are indexed by -element subsets of and , respectively, in their lexic ...
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Vandermonde Matrix
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an matrix :V=\begin 1 & x_1 & x_1^2 & \dots & x_1^\\ 1 & x_2 & x_2^2 & \dots & x_2^\\ 1 & x_3 & x_3^2 & \dots & x_3^\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ 1 & x_m & x_m^2 & \dots & x_m^ \end, or :V_ = x_i^ \, for all indices and . Some authors define the Vandermonde matrix as the transpose of the above matrix. The determinant of a square Vandermonde matrix is called a ''Vandermonde polynomial'' or ''Vandermonde determinant''. Its value is the polynomial :\det(V) = \prod_ (x_j - x_i) which is non-zero if and only if all x_i are distinct. The Vandermonde determinant was sometimes called the ''discriminant'', although, presently, the discriminant of a polynomial is the square of the Vandermonde determinant of the roots of the polynomial. The Vandermonde determinant is an alternating form in the x_i, meaning that exchang ...
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Pólya Frequency Functions
Pólya (Hungarian for "swaddling clothes") is a surname. People with the surname include: * Eugen Alexander Pólya (1876-1944), Hungarian surgeon, elder brother of George Pólya ** Reichel-Polya Operation, a type of partial gastrectomy developed by Eugen Pólya and Friedrich Paul Reichel * George Pólya (1887-1985), Hungarian mathematician ** Pólya Prize (LMS), awarded by the London Mathematical Society ** Pólya Prize (SIAM), awarded by the Society for Industrial and Applied Mathematics ** Pólya Award, awarded by the Mathematical Association of America (MAA) ** Pólya enumeration theorem ** Pólya conjecture ** Hilbert–Pólya conjecture ** Pólya–Szegő inequality ** Multivariate Pólya distribution ** The Pólya–Vinogradov inequality * (1886-1937), Hungarian graphic artist See also *polyA Polyadenylation is the addition of a poly(A) tail to an RNA transcript, typically a messenger RNA (mRNA). The poly(A) tail consists of multiple adenosine monophosphates; in oth ...
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Variation Diminishing Property
In mathematics, the variation diminishing property of certain mathematical objects involves diminishing the number of changes in sign (positive to negative or vice versa). Variation diminishing property for Bézier curves The variation diminishing property of Bézier curves is that they are smoother than the polygon formed by their control points. If a line is drawn through the curve, the number of intersections with the curve will be less than or equal to the number of intersections with the control polygon. In other words, for a Bézier curve ''B'' defined by the control polygon P, the curve will have no more intersection with any plane as that plane has with P. This may be generalised into higher dimensions. This property was first studied by Isaac Jacob Schoenberg in his 1930 paper, . He went on to derive it by a transformation of Descartes' rule of signs. Proof The proof uses the process of repeated degree elevation of Bézier curve#Repeated degree elevation, Bézier c ...
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Green's Function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential operator, then * the Green's function G is the solution of the equation \operatorname G = \delta, where \delta is Dirac's delta function; * the solution of the initial-value problem \operatorname y = f is the convolution (G \ast f). Through the superposition principle, given a linear ordinary differential equation (ODE), \operatorname y = f, one can first solve \operatorname G = \delta_s, for each , and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of . Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are s ...
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Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ...
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Kernel (algebra)
In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the ''null space'', is the kernel of the linear map defined by the matrix. The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.See and . For some types of structure, such as abelian groups and vector spaces, the possible kernels are exactly the substructures of the same type. This is not always the case, and, sometimes, the possible kernels have received a special name, such as normal subgroup for groups and two-sided ideals for r ...
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Positive-definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines a ...
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Positive-definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines a ...
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