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Matrix Sign Function
In mathematics, the matrix sign function is a matrix function on square matrices analogous to the complex sign function. It was introduced by J.D. Roberts in 1971 as a tool for model reduction and for solving Lyapunov and Algebraic Riccati equation in a technical report of Cambridge University, which was later published in a journal in 1980. Definition The matrix sign function is a generalization of the complex signum function \operatorname(z)= \begin 1 & \text \mathrm(z) > 0, \\ -1 & \text \mathrm(z) < 0, \end to the matrix valued analogue . Although the sign function is not analytic, the is well defined for all matrices that have no |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Left-half Plane
A midfielder is an outfield position in association football. Midfielders may play an exclusively defensive role, breaking up attacks, and are in that case known as defensive midfielders. As central midfielders often go across boundaries, with mobility and passing ability, they are often referred to as deep-lying midfielders, play-makers, box-to-box midfielders, or holding midfielders. There are also attacking midfielders with limited defensive assignments. The size of midfield units on a team and their assigned roles depend on what formation is used; the unit of these players on the pitch is commonly referred to as the midfield. Its name derives from the fact that midfield units typically make up the in-between units to the defensive units and forward units of a formation. Managers frequently assign one or more midfielders to disrupt the opposing team's attacks, while others may be tasked with creating goals, or have equal responsibilities between attack and defence. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consistent And Inconsistent Equations
In mathematics and particularly in algebra, a system of equations (either linear or nonlinear) is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold true as an identity. In contrast, a linear or non linear equation system is called inconsistent if there is no set of values for the unknowns that satisfies all of the equations. If a system of equations is inconsistent, then it is possible to manipulate and combine the equations in such a way as to obtain contradictory information, such as , or x^3 + y^5 = 5 and x^3 + y^3 = 6 (which implies ). Both types of equation system, consistent and inconsistent, can be any of overdetermined (having more equations than unknowns), underdetermined (having fewer equations than unknowns), or exactly determined. Simple examples Underdetermined and consistent The system :\begin x+y+z &= 3, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Overdetermined System
In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients. However, an overdetermined system will have solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others. The terminology can be described in terms of the concept of constraint counting. Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom. Therefore, the critical case occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint. The ''overdetermined'' case occurs when the system has been overconstrained — that is, when the equations outnu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although note that in quantum mechanics, A^\ast typically means the complex conjugate only, and not the conjugate transpose. Alternative characterizations Her ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester Equation
In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form: :A X + X B = C. Then given matrices ''A'', ''B'', and ''C'', the problem is to find the possible matrices ''X'' that obey this equation. All matrices are assumed to have coefficients in the complex numbers. For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size. But more generally, ''A'' and ''B'' must be square matrices of sizes ''n'' and ''m'' respectively, and then ''X'' and ''C'' both have ''n'' rows and ''m'' columns. A Sylvester equation has a unique solution for ''X'' exactly when there are no common eigenvalues of ''A'' and −''B''. More generally, the equation ''AX'' + ''XB'' = ''C'' has been considered as an equation of bounded operators on a (possibly infinite-dimensional) Banach space. In this case, the condition for the uniqueness of a solution ''X'' is almost ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable 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 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 its roots have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Günther Schulz (mathematician)
Günther, Guenther, Ginther, Gunther, and the variants Günter, Guenter, Guenther, Ginter, and Gunter, are Germanic names derived from ''Gunthere, Gunthari'', composed of '' *gunþiz'' "battle" (Old Norse ''gunnr'') and ''heri, hari'' "army". Gunder and Gunnar are the North Germanic equivalents in Scandinavia. The name may refer to: People * Günther (given name) * Günther (singer), the stage persona of Swedish musician Mats Söderlund * Günther (surname) Places *Gunther Island, in Humboldt Bay, California Ships *, a number of ships with this name Fictional characters * Gunther, a character in the television show ''Friends'' * Gunther, mayor of the city of Motril in the video game ''Grand Theft Auto V'' * Gunther, a character in '' Kick Buttowski: Suburban Daredevil'' * Günther Bachmann, a character in the film ''A Most Wanted Man'' * Gunther Berger, a character in the '' Luann'' comic strip * Gunther Breech, a character in the Canadian animated TV show '' Jane and the Dr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of A Matrix
In mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix is said to be a square root of if the matrix product is equal to . Some authors use the name ''square root'' or the notation only for the specific case when is positive semidefinite, to denote the unique matrix that is positive semidefinite and such that (for real-valued matrices, where is the transpose of ). Less frequently, the name ''square root'' may be used for any factorization of a positive semidefinite matrix as , as in the Cholesky factorization, even if . This distinct meaning is discussed in '. Examples In general, a matrix can have several square roots. In particular, if A = B^2 then A=(-B)^2 as well. The 2×2 identity matrix \textstyle\begin1 & 0\\ 0 & 1\end has infinitely many square roots. They are given by :\begin \pm 1 & 0\\ 0 & \pm 1\end and \begin a & b\\ c & -a\end where (a, b, c) are any numbers (real or complex) such that a^2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
