Routh–Hurwitz Matrix

	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]
picture info	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]
picture info	Square Matrix In mathematics, a square matrix is a Matrix (mathematics), 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 used to represent simple linear transformations, such as Shear mapping, shearing or Rotation (mathematics), rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and \mathbf is a column vector describing the Position (vector), position of a point in space, the product R\mathbf yields another column vector describing the position of that point after that rotation. If \mathbf is a row vector, the same transformation can be obtained using where R^ is the transpose of Main diagonal The entries a_ () form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, mathematical analysis, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died in Zürich, in Switzerland. His father Salomon Hurwitz, a merchant, was not wealthy. Hurwitz's mother, Elise Wertheimer, died when he was three years old. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed except for an older brother, Julius, with whom he developed an arithmetical theory for complex continued fractions circa 1890. Hurwitz entered the in Hildesheim in 1868. He was taught mathematics there by Hermann Schubert. Schubert persuaded Hurwitz's father to allow him to attend university, and arranged for Hurwitz to study with Felix Klein at Munich. Salomon Hurwitz could not afford to send his son to university, but his friend, Mr. Edwards, assisted financially. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Stable Polynomial In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either: * all its roots lie in the open left half-plane, or * all its roots lie in the open unit disk. The first condition provides stability for continuous-time linear systems, and the second case relates to stability of discrete-time linear systems. A polynomial with the first property is called at times a Hurwitz-stable polynomial and with the second property a Schur-stable polynomial. Stable polynomials arise in control theory and in mathematical theory of differential and difference equations. A linear, time-invariant system (see LTI system theory) is said to be BIBO stable if every bounded input produces bounded output. A linear system is BIBO stable if its characteristic polynomial is stable. The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. In practic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Minor (linear Algebra) In linear algebra, a minor of a matrix (mathematics), matrix is the determinant of some smaller square matrix generated from by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and Inverse matrix, inverse of square matrices. The requirement that the square matrix be smaller than the original matrix is often omitted in the definition. Definition and illustration First minors If is a square matrix, then the ''minor'' of the entry in the -th row and -th column (also called the ''minor'', or a ''first minor'') is the determinant of the submatrix formed by deleting the -th row and -th column. This number is often denoted . The ''cofactor'' is obtained by multiplying the minor by . To illustrate these definitions, consider the following matrix, \begin 1 & 4 & 7 \\ 3 & 0 & 5 \\ -1 & 9 & 11 \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hurwitz Determinant In mathematics, Hurwitz determinants were introduced by , who used them to give a criterion for all roots of a polynomial to have negative real part. Definition Consider a characteristic polynomial ''P'' in the variable ''λ'' of the form: : P(\lambda)= a_0 \lambda^n + a_1 \lambda^ + \cdots + a_ \lambda + a_n where a_i, i=0,1,\ldots,n, are real. The square Hurwitz matrix associated to ''P'' is given below: : 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. The ''i-''th ''Hurwitz determinant'' is the ''i-''th leading principal minor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Characteristic Polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the correspondi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hurwitz-stable Matrix In mathematics, a Hurwitz-stable matrix, or more commonly simply Hurwitz matrix, is a square matrix whose eigenvalues all have strictly negative real part. Some authors also use the term stability matrix. Such matrices play an important role in control theory. Definition A square matrix A is called a Hurwitz matrix if every eigenvalue of A has strictly negative real part, that is, :\operatorname lambda_i< 0\, for each eigenvalue $\lambda_i$ . $A$ is also called a stable matrix, because then the differential equation : $\dot x = A x$ is asymptotically stable , that is, $x(t)\to 0$ as $t\to\infty.$ If $G(s)$ is a (matrix-valued) [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Routh–Hurwitz Stability Criterion In the control theory, control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stable polynomial, stability of a linear time-invariant system, linear time-invariant (LTI) dynamical system or control system. A stability theory, stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as time goes on. The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in 1876 to determine whether all the root of a function, roots of the characteristic polynomial of a linear system have negative real parts. German mathematician Adolf Hurwitz independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the Routh–Hurwitz matrix, Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Liénard–Chipart Criterion In control theory, the Liénard–Chipart criterion is a stability criterion modified from the Routh–Hurwitz stability criterion, proposed in 1914 by French physicists A. Liénard and M. H. Chipart. This criterion has a computational advantage over the Routh–Hurwitz criterion because it involves only about half the number of determinant computations. Algorithm The Routh–Hurwitz stability criterion says that a necessary and sufficient condition for all the roots of the polynomial with real coefficients f(z) = a_0 z^n + a_1 z^ + \cdots + a_n, \quad a_0 > 0 to have negative real parts (i.e. is Hurwitz stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...) is that \Delta_1 > 0,\, \Delta_2 > 0, \ \ldots, \ \Delta_n > 0, where is the -th leading principal minor of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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 $\$ , $u_i \neq 0$ , $i = 1,...,n$ are the eigenvalues of an -dimensional $P_0$ -matrix, then :: $, \arg(u_i), \leq \pi - \frac,\ i = 1,...,n$ 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]
	SIAM Journal On Applied Mathematics The ''SIAM Journal on Applied Mathematics'' is a peer-reviewed academic journal in applied mathematics published by the Society for Industrial and Applied Mathematics (SIAM), with Paul A. Martin (Colorado School of Mines) as its editor-in-chief. It was founded in 1953 as SIAM's first journal, the ''Journal of the Society for Industrial and Applied Mathematics'', and was given its current name in 1966. In most years since 1999, it has been ranked by SCImago Journal Rank as a second-quartile journal in applied mathematics. Together with ''Communications on Pure and Applied Mathematics ''Communications on Pure and Applied Mathematics'' is a monthly peer-reviewed scientific journal which is published by John Wiley & Sons on behalf of the Courant Institute of Mathematical Sciences. It covers research originating from or solicited ...'' it has been called "one of the two greatest American entries in applied math".. References {{Society for Industrial and Applied Mathematics Applied ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

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