R.A. Horn and C. Johnson: Matrix analysis. Cambridge University Press, Cambridge, U.K., 1985. Home/Search Document Not in Database Summary Related Articles Check |
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Deriving Sufficient Conditions for Global Asymptotic Stability of .. - Qi, Qi (2003) (Correct)
....if B I 2 P for any 0, where I is the identity matrix. For any matrix B 2 2 (B) 2 max (B B ) i.e. 2 (B) is the largest eigenvalue of the symmetric part of B. All the mathematical facts concerning the eigenvalues of a matrix used in this paper can be found in the book [17] 2 Nonsmooth Analysis on Lipschitz Functions In this section, we rst review some concepts which are essential for conducting nonsmooth analysis on Lipschitz functions. Among those is the core concept of the Generalized Jacobian, to which we shall pay more attention since its calculation plays ....
R.A. Horn and C.A. Johnson Matrix Analysis, Cambridge University Press, 1985.
Channel Estimation for Space-Time Orthogonal Block Codes - Budianu, Tong (2002) (3 citations) (Correct)
....the matrix l( y) anymore. Instead, we can write, F( y) and then go,go q gl,go q g2 are the eigenvalues of (7) Thus we have 0 (7 ) 7 ) 96) Since the matrix , 1 is tall and full column rank for any positive definite matrix U the matrix JUJ1 is also positive definite (see [14]) The formula in the theorem follows from (96) We need to how the FIM matrix for the horizontal power allocation scheme described below varies with b: ql ZX [1 q , 0, O] T, yl[O, 1, 1] T, 97) q52 = o, o] r, r2= 7, L. r. 98) Like in the proof of theorem 6, consider ....
R.A. Horn and C.R. Johnson Matrix Analysis, Cambridge University Press, 1985.
Faster MCMC Estimation Along One Walk - Niemiro, Pokarowski (Correct)
....when restricted to the subspace orthogonal to constant functions. 7. Lemma. If hf; 1i = 0 then jjP t f jj t jjf jj. Proof. By reversibility, we have jjP f jj 2 = hPf; Pfi = hf; P 2 fi: Now, use the well known minimax characterization of eigenvalues of a selfadjoint operator (c.f. [4], page 176, e.g. Since v 1 1, the second largest eigenvalue of P 2 is 2 = max f :hf;1i=0 hf; P 2 fi=hf; fi: But the eigenvalues of P 2 are 1; 2 2 ; 2 s , hence 2 = max( 2 2 ; 2 s ) Thus for hf; 1i = 0 we have jjP f jj 2 2 jjf jj 2 . To obtain the ....
R. Horn, C. Johnson: Matrix Analysis, Cambridge Univ. Press, 1985.
Complex Symmetric Eigenproblems - Gansterer, Haunschmid, Ueberhuber (2000) (Correct)
....Vienna christof aurora.tuwien.ac.at April 2000 AURORA TR2000 07 Abstract This report is focused on complex symmetric (non Hermitian) eigenproblems. Such problems do not occur as frequently in practice as real symmetric or complex Hermitian problems (see, for example, Horn and Johnson [17], Craven [8] Ohnami amd Mikami [22] Bar on and Ryaboy [3] Complex symmetric eigenproblems arise, for example, in applications dealing with resonance scattering in quantum mechanics. Details are given of an application dealing with the accurate calculation of atomic resonances near metal ....
....(Stewart [23] Wilkinson [26] Demmel, K agstr om [10] The mathematical properties of complex symmetric matrices are quite different from those of real symmetric or complex Hermitian matrices. Since any general complex matrix is similar to some complex symmetric matrix (Horn and Johnson [17]) it seems as if this special structure would be of no particular signi cance. The standard approach to solving complex symmetric eigenproblems (in particular, the only approach available in Lapack [1] is to treat them as unsymmetric eigenproblems. This means, that complex symmetric matrices are ....
R. A. Horn, C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge, 1985.
A Divide-and-Conquer Method for Symmetric Banded.. - Gansterer, Schneid.. (1998) (Correct)
....is required (Dhillon [10] and therefore the former approach will be pursued in this report. In this chapter it will be shown how an orthonormal basis for the eigenspace of S can be constructed. Since eigenvectors corresponding to distinct eigenvalues are orthogonal (Horn and Johnson [19]) only an orthonormal basis of the eigenspace corresponding to one given eigenvalue i has to be constructed. Three cases have to be distinguished. Case I In the rst case assume that i = 2 ( and has an algebraic multiplicity l, 1 l b. Since i is an eigenvalue of S, the ....
....to i . They are orthonormal by construction. It remains to be shown that all l eigenvectors corresponding to the eigenvalue i have been found, that is, m = l. First, since the geometric multiplicity of an eigenvalue is less or equal its algebraic multiplicity (Horn and Johnson [19]) m l. Equality can be shown indirectly. Assume that m l. Then there must be another eigenvector z i of S with kz i k 2 = 1, z i Z i = 0, and, according to (4.1) UU i I n z i = 0 (4.7) I b U ( i I n ) 1 U U z i = 0: This implies that U z i ....
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R. A. Horn, C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge, 1985.
Structured Jordan Canonical Forms for Structured Matrices that .. - Mehrmann, Xu (1999) (Correct)
.... as in (2) A matrix E 2 C j Thetak satisfies N j E = EN k if and only if E has the form E = 8 : T j = k; h 0 T i j k; T 0 # j k; 8) where T has the form (7) For more complicated nilpotent matrices in Jordan form we have the following well known Lemma, see [7, 10], where we denote the set of j Theta k rectangular upper triangular Toeplitz matrices E as in (8) by G j Thetak . Lemma 3 Let N be a nilpotent Jordan matrix of the form N = diag(N r 1 ; N rs ) A matrix E commutes with N if and only if E has the block structure E = E i;j ] s Thetas , ....
R.A. Horn and C.R. Johnson Matrix Analysis, Cambridge University Press, 1985.
The Convex Analysis of Unitarily Invariant Matrix Functions - Lewis (1995) (Correct)
....apparent. Many of the most important examples of norms k Delta k on I C m Thetan are unitarily invariant: kV XUk = kXk for any X 2 I C m Thetan ; U 2 U n ; V 2 Um (where U n denotes the set of n Theta n unitary matrices) A famous fundamental result of von Neumann s [9] see for example [5]) states that such matrix norms can be characterized as composite functions of the form f(oe( Delta) f ffi oe, where the function oe : I C m Thetan IR q (with q = minfm;ng) has components oe 1 (X) oe 2 (X) oe q (X) 0, the singular values of the matrix X, and the function f : ....
R.A. Horn and C. Johnson: Matrix analysis. Cambridge University Press, Cambridge, U.K., 1985.
On Extreme Singular Values of Matrix Valued Functions - Qi, Womersley (1996) (Correct)
....value of a matrix is a convex function of the elements of the matrix. Identifying the elements which achieve the maximum gives a concise formula for the subdifferential (generalized gradient) of the sum of the largest singular values. Bounds on singular values of matrices are widely known (see [14], 16] for example) Subramani [30] provides a review of the inequalities relevant to characterizing sums of the largest singular values of a rectangular matrix. Another convexity property of the largest singular value is given by Sezinger and Overton [29] They establish that the largest singular ....
....1 i 1 i 2 : i r p. For a matrix D = diag(d 11 ; d pp ) 2 IR m Thetan , all the off diagonal elements of D are zero. For a symmetric matrix A, we use ae(A) to denote its spectral radius. The positive semi definite partial ordering on S n is used to express matrix inequalities [14]. Thus for A; B 2 S n the inequality A B means that A Gamma B is positive semi definite. The Frobenius inner product hA; Bi of two matrices A; B 2 IR m Thetan is hA; Bi = tr(AB T ) m X i=1 n X j=1 a ij b ij : For any nonsingular matrices E 2 IR m Thetam and F 2 IR n Thetan , ....
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R. A. Horn and C. R. Johnson: Matrix Analysis, Cambridge University Press, Cambridge, 1985.
Journal of Convex Analysis Volume 2 (1995), No.1/2.. - Matrix Functions Lewis (Correct)
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R.A. Horn and C. Johnson: Matrix analysis. Cambridge University Press, Cambridge, U.K., 1985.
Complex Symmetric - Eigenproblems Wilfried Gansterer (2000) (Correct)
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R. A. Horn, C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge, 1985.
Unknown - Divide-And-Conquer Method For (1999) (Correct)
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R. A. Horn, C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge, 1985.
Mathematical Properties of Equilibrium Systems - Gansterer, Schneid, Ueberhuber (2003) (Correct)
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R. A. Horn, C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge, 1985.
A Survey of Equilibrium Systems - Gansterer, Schneid, Ueberhuber (2002) (Correct)
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R. A. Horn, C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge, 1985.
A REDUCE package for the computation of several matrix normal forms - Rebbeck (1994) (Correct)
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Roger A. Horn and Charles A. Johnson: Matrix Analysis. Cambridge University Press (1990)
Semidefinite Programs: New Search Directions, Smoothing-Type.. - Kanzow, Nagel (Correct)
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R.A. Horn and C.R. Johnson: Matrix Analysis. Cambridge University Press, 1985.
Semidefinite Programs: New Search Directions, Smoothing-Type.. - Kanzow, Nagel (2001) (Correct)
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R.A. Horn and C.R. Johnson: Matrix Analysis. Cambridge University Press, 1985.
On Independent Component Analysis for Multimedia Signals - Hansen, Larsen, Kolenda (2000) (Correct)
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R.A. Horn & C.R. Johnson: Matrix Analysis, Cambridge University Press: Cambridge UK, 1994.
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