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The classical singular value decomposition for a matrix A∈Cm×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA∗ and A∗A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices and . More generally, we consider the matrix triple , where are invertible and either complex symmetric or complex skew-symmetric, and we provide a canonical form under transformations of the form , where X,Y are nonsingular. 相似文献
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For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B--XF--F+X+XB+X=0, where , and with a nonnegative matrix P, positive diagonal matrices D±, and nonnegative parameters f, and . We prove the existence of the minimal nonnegative solution X∗ under the physically reasonable assumption , and study its numerical computation by fixed-point iteration, Newton’s method and doubling. We shall also study several special cases; e.g. when and P is low-ranked, then is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results. 相似文献
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Bo Hou 《Linear algebra and its applications》2011,435(8):1987-1996
Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider an ordered pair of F-linear transformations A:V→V and A∗:V→V that satisfy the following conditions: (i) each of A,A∗ is diagonalizable on V; (ii) there exists an ordering of the eigenspaces of A such that A∗Vi⊆V0+V1+?+Vi+1 for 0?i?d, where V-1:=0 and Vd+1:=0; (iii) there exists an ordering of the eigenspaces of A∗ such that for 0?i?δ, where and . We call such a pair a Hessenberg pair on V. It is known that if the Hessenberg pair A,A∗ on V is irreducible then d=δ and for 0?i?d the dimensions of Vi and coincide. We say a Hessenberg pair A,A∗ on V is sharp whenever it is irreducible and .In this paper, we give the definitions of a Hessenberg system and a sharp Hessenberg system. We discuss the connection between a Hessenberg pair and a Hessenberg system. We also define a finite sequence of scalars called the parameter array for a sharp Hessenberg system, which consists of the eigenvalue sequence, the dual eigenvalue sequence and the split sequence. We calculate the split sequence of a sharp Hessenberg system. We show that a sharp Hessenberg pair is a tridiagonal pair if and only if there exists a nonzero nondegenerate bilinear form on V that satisfies 〈Au,v〉=〈u,Av〉 and 〈A∗u,v〉=〈u,A∗v〉 for all u,v∈V. 相似文献
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Zhichun Zhai 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(8):2611-2630
Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤p≤q<∞, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v0(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,∞). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality. 相似文献
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Let s∈R. In this paper, the authors first establish the maximal function characterizations of the Besov-type space with and τ∈[0,∞), the Triebel-Lizorkin-type space with p∈(0,∞), q∈(0,∞] and τ∈[0,∞), the Besov-Hausdorff space with p∈(1,∞), q∈[1,∞) and and the Triebel-Lizorkin-Hausdorff space with and , where t′ denotes the conjugate index of t∈[1,∞]. Using this characterization, the authors further obtain the local mean characterizations of these function spaces via functions satisfying the Tauberian condition and establish a Fourier multiplier theorem on these spaces. All these results generalize the existing classical results on Besov and Triebel-Lizorkin spaces by taking τ=0 and are also new even for Q spaces and Hardy-Hausdorff spaces. 相似文献
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