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Quadratic Eigenvalue Problem
In mathematics, the quadratic eigenvalue problemF. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235–286. (QEP), is to find scalar eigenvalues \lambda, left eigenvectors y and right eigenvectors x such that : Q(\lambda)x = 0 ~ \text ~ y^\ast Q(\lambda) = 0, where Q(\lambda)=\lambda^2 M + \lambda C + K, with matrix coefficients M, \, C, K \in \mathbb^ and we require that M\,\neq 0, (so that we have a nonzero leading coefficient). There are 2n eigenvalues that may be ''infinite'' or finite, and possibly zero. This is a special case of a nonlinear eigenproblem. Q(\lambda) is also known as a quadratic polynomial matrix. Spectral theory A QEP is said to be regular if \text (Q(\lambda)) \not \equiv 0 identically. The coefficient of the \lambda^ term in \text(Q(\lambda)) is \text(M), implying that the QEP is regular if M is nonsingular. Eigenvalues at infinity and eigenvalues at 0 may be exchanged by considering the reversed polynomial, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Stiffness Matrix
In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation. The stiffness matrix for the Poisson problem For simplicity, we will first consider the Poisson problem : -\nabla^2 u = f on some domain , subject to the boundary condition on the boundary of . To discretize this equation by the finite element method, one chooses a set of '' basis functions'' defined on which also vanish on the boundary. One then approximates : u \approx u^h = u_1\varphi_1+\cdots+u_n\varphi_n. The coefficients are determined so that the error in the approximation is orthogonal to each basis function : : \int_ \varphi_i\cdot f \, dx = -\int_ \varphi_i\nabla^2u^h \, dx = -\sum_j\left(\int_ \varphi_i\nabla^2\varphi_j\,dx\right)\, u_j = \sum_j\left(\int_ \nabla\varphi_i\cdot\nabla\varphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Hamiltonian Matrix
In mathematics, a Hamiltonian matrix is a -by- matrix such that is symmetric, where is the skew-symmetric matrix :J = \begin 0_n & I_n \\ -I_n & 0_n \\ \end and is the -by- identity matrix. In other words, is Hamiltonian if and only if where denotes the transpose.. (Not to be confused with Hamiltonian (quantum mechanics)) Properties Suppose that the -by- matrix is written as the block matrix : A = \begin a & b \\ c & d \end where , , , and are -by- matrices. Then the condition that be Hamiltonian is equivalent to requiring that the matrices and are symmetric, and that .. Another equivalent condition is that is of the form with symmetric. It follows easily from the definition that the transpose of a Hamiltonian matrix is Hamiltonian. Furthermore, the sum (and any linear combination) of two Hamiltonian matrices is again Hamiltonian, as is their commutator. It follows that the space of all Hamiltonian matrices is a Lie algebra, denoted . The dimension of is . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Identity Matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end. The term unit matrix has also been widely used, but the term ''identity matrix'' is now standard. The term ''unit matrix'' is ambiguous, because it is also used for a matrix of on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Companion Matrix
In linear algebra, the Frobenius companion matrix of the monic polynomial p(x)=c_0 + c_1 x + \cdots + c_x^ + x^n is the square matrix defined as C(p)=\begin 0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & \dots & 0 & -c_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_ \end. Some authors use the transpose of this matrix, C(p)^T , which is more convenient for some purposes such as linear recurrence relations ( see below). C(p) is defined from the coefficients of p(x), while the characteristic polynomial as well as the minimal polynomial of C(p) are equal to p(x) . In this sense, the matrix C(p) and the polynomial p(x) are "companions". Similarity to companion matrix Any matrix with entries in a field has characteristic polynomial p(x) = \det(xI - A) , which in turn has companion matrix C(p) . These matrices are related as follows. The following statements are equivalent: * ''A'' is similar over ''F'' to C(p) , i.e. ''A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Matrix Pencil
In linear algebra, a matrix pencil is a matrix-valued polynomial function defined on a field K, usually the real or complex numbers. Definition Let K be a field (typically, K \in \; the definition can be generalized to rngs), let \ell \ge 0 be a non-negative integer, let n > 0 be a positive integer, and let A_0, A_1, \dots, A_\ell be n\times n matrices (i. e. A_i \in \mathrm(K, n \times n) for all i = 0, \dots, \ell). Then the matrix pencil defined by A_0, \dots, A_\ell is the matrix-valued function L \colon K \to \mathrm(K, n \times n) defined by :L(\lambda) = \sum_^\ell \lambda^i A_i. The ''degree'' of the matrix pencil is defined as the largest integer 0 \le k \le \ell such that A_k \ne 0, the n \times n zero matrix over K. Linear matrix pencils A particular case is a linear matrix pencil L(\lambda) = A - \lambda B (where B \ne 0). We denote it briefly with the notation (A, B), and note that using the more general notation, A_0 = A and A_1 = -B (not B). Proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Matrix Polynomial
In mathematics, a matrix polynomial is a polynomial with square matrices as variables. Given an ordinary, scalar-valued polynomial : P(x) = \sum_^n =a_0 + a_1 x+ a_2 x^2 + \cdots + a_n x^n, this polynomial evaluated at a matrix A is :P(A) = \sum_^n =a_0 I + a_1 A + a_2 A^2 + \cdots + a_n A^n, where I is the identity matrix. Note that P(A) has the same dimension as A. A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices ''A'' in a specified matrix ring ''Mn''(''R''). Matrix polynomials are often demonstrated in undergraduate linear algebra classes due to their relevance in showcasing properties of linear transformations represented as matrices, most notably the Cayley–Hamilton theorem. Characteristic and minimal polynomial The characteristic polynomial of a matrix ''A'' is a scalar-valued polynomial, defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Schur Decomposition
In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily similar to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix. Statement The complex Schur decomposition reads as follows: if is an square matrix with complex entries, then ''A'' can be expressed as (Section 2.3 and further at p. 79(Section 7.7 at p. 313 A = Q U Q^ for some unitary matrix ''Q'' (so that the inverse ''Q''−1 is also the conjugate transpose ''Q''* of ''Q''), and some upper triangular matrix ''U''. This is called a Schur form of ''A''. Since ''U'' is similar to ''A'', it has the same spectrum, and since it is triangular, its eigenvalues are the diagonal entries of ''U''. The Schur decomposition implies that there exists a nested sequence of ''A''-invariant subspaces , and that there exists an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Schur Form
In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation, named after Issai Schur, is a matrix decomposition. It allows one to write an arbitrary complex square matrix as unitarily similar to an upper triangular matrix whose diagonal elements are the eigenvalues of the original matrix. Statement The complex Schur decomposition reads as follows: if is an square matrix with complex entries, then ''A'' can be expressed as (Section 2.3 and further at p. 79(Section 7.7 at p. 313 A = Q U Q^ for some unitary matrix ''Q'' (so that the inverse ''Q''−1 is also the conjugate transpose ''Q''* of ''Q''), and some upper triangular matrix ''U''. This is called a Schur form of ''A''. Since ''U'' is similar to ''A'', it has the same spectrum, and since it is triangular, its eigenvalues are the diagonal entries of ''U''. The Schur decomposition implies that there exists a nested sequence of ''A''-invariant subspaces , and that there exists an ordere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Generalized Eigenvalue Problem
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Fundamental theory of matrix eigenvectors and eigenvalues A (nonzero) vector of dimension is an eigenvector of a square matrix if it satisfies a linear equation of the form \mathbf \mathbf = \lambda \mathbf for some scalar . Then is called the eigenvalue corresponding to . Geometrically speaking, the eigenvectors of are the vectors that merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. The above equation is called the eigenvalue equation or the eigenvalue problem. This yields an equation for the eigenvalues p\left(\lambda\right) = \det\lef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Fluid Dynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion) and (the study of water and other liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moment (physics), moments on aircraft, determining the mass flow rate of petroleum through pipeline transport, pipelines, weather forecasting, predicting weather patterns, understanding nebulae in interstellar space, understanding large scale Geophysical fluid dynamics, geophysical flows involving oceans/atmosphere and Nuclear weapon design, modelling fission weapon detonation. Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Damping Matrix
In applied mathematics, a damping matrix is a matrix corresponding to any of certain systems of linear ordinary differential equations. A damping matrix is defined as follows. If the system has ''n'' degrees of freedom ''u''''n'' and is under application of ''m'' damping forces. Each force can be expressed as follows: : f_=c_ \dot+c_ \dot+\cdots+c_ \dot=\sum_^n c_\dot It yields in matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ... form; : F_D=C \dot where C is the damping matrix composed by the damping coefficients: : C=(c_)_ Mechanical engineering Classical mechanics {{mathapplied-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |