A. Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math., 76:620--630, 1954. Home/Search Document Not in Database Summary Related Articles Check |
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The Mathematics Of Eigenvalue Optimization - Lewis (2003) (1 citation) (Correct)
....orthogonal projection (under the Killing form) onto a of any orbit of K acting on p is the convex hull of an orbit of the Weyl group. In our example, the orthogonal projection is the map that sets the o diagonal entries of a trace zero symmetric matrix to zero, so we deduce the following result [30]. The trace zero assumption below is easily dispensable. Corollary 5.2 (Horn, 1954) The diagonals of all symmetric matrices similar to a given trace zero symmetric matrix X form a polytope. This polytope is the convex hull of all vectors with entries permutations of the eigenvalues of X. What ....
R.A. Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. American Journal of Mathematics, 76:620-630, 1954.
Concavity of Monotone Matrix Functions of Finite Order - Roy Mathias Department (1990) (Correct)
....= diag( A) There exist scalars i i (A) i = 1; n such that k X i=1 i k X i=1 i (B) k = 1; n Gamma 1 (3.4) n X i=1 i = n X i=1 i (B) 3.5) see, e.g. the proof of Theorem 7.4.45 in [7] In view of (3.4) and (3.5) it follows from a theorem of A. Horn ([6], see also [7, Theorem 4.3.32] that there is a unitary matrix V such that (V BV ) ii = i . Let be a primitive n th root of unity, and define W j diag( 1; n Gamma1 ] Then we have UAU = diag( A) diag( 1 n n X i=1 W i (V BV ) W i ) If we choose U i = ....
A. Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math., 76:620--630, 1954.
Trust Region Problems And Nonsymmetric Perturbation Theory - Stern, Wolkowicz (1993) (1 citation) (Correct)
....j 2 k = f(fl k ) g 0 (fl k ) k = 1; 2; n; and t = n X i=1 ffi i Gamma n Gamma1 X i=1 fl i : Then the eigenvalues of A are the roots of f( Delta) which are the n values ffi i . A modification of the proof yields the case of nondistinct ffi i ) 2 Remark 4.3 A. Horn [10] proved the following inverse eigenvalue result, which may be viewed as a converse to Theorem 4.4: If we are given real numbers a 11 a 22 : a nn and ffi 1 ffi 2 : ffi n such that the majorization (4.33) holds, then there exists an n Theta n real symmetric matrix A with diagonal ....
A. Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math., 76:620--630, 1954.
Convexity of the Joint Numerical Range - Li, Poon (1998) (Correct)
....n X j=1 z j = 1; z j 0 for j = 1; n: Moreover, for j = 1; n Gamma 1, 4jz n j j 2 = jx n j y n j j 2 ( p x j x j 1 p y j y j 1 ) 2 (x j y j ) x j 1 y j 1 ) Hence z 2 W 1 (A) Since W 1 (A) is closed, we conclude that it is a convex set. 2 A result of Horn [8] implies that W 1 (E 11 ; E 22 ; E nn ) is convex. Theorem 3.1 strengthens this statement. Note that a maximal linearly independent convex family of the first joint numerical range may not have 2(n Gamma 1) 1 elements as shown in the following example. Example 3.2 Suppose (k; n) 1; ....
....J k K) K = GammaK t g = ae 0 X X t 0 : X 2 R k Theta(n Gammak) oe ; which has dimension k(n Gamma k) Consequently, if A = A 1 ; Am ) such that dim (span fI; A 1 ; Amg) k(n Gamma k) 1; then W k (A) is not convex. By this theorem and a result of Horn [8], we have the following corollary. Corollary 5.3 The elements of A = E 11 ; E 22 ; E nn ) form a maximal linearly independent convex family for W 1 (A) Rectangular matrices Suppose A = A 1 ; Am ) where A 1 ; Am are n Theta r matrices over F = R or C. To be specific, ....
A. Horn, Doubly Stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math. 76 (1954), 620-630.
Gradient Flow Methods For Matrix Completion With - Prescribed Eigenvalues Moody (Correct)
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A. Horn. Doubly stochastic matrices and the diagonal of a rotation matrix. Amer. J. Math., 76:620--630, 1954.
Constructing a Hermitian Matrix from Its - Diagonal Entries And (Correct)
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A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math., 76(1956), 620-630.
On Constructing Matrices With Prescribed - Singular Values And (Correct)
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A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math., 76(1956), 620-630.
A Fast Recursive Algorithm For - Constructing Matrices With (Correct)
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A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math., 76(1956), 620-630.
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