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Mathematics & Statistics Auburn University, Alabama, USA On Mathematics & Statistics Auburn University, Alabama, USA May 25, 2011 Home Page On Bertram Kostant’s paper: On convexity, the Title Page Weyl group and the Iwasawa decomposition Contents Ann. Sci. Ecole´ Norm. Sup. (4) 6 (1973), 413–455 JJ II J I Page 1 of 30 Tin-Yau Tam Go Back Full Screen 2011 AU Summer Seminar Close Quit [email protected] 1. Some matrix results Notation: Given A ∈ Cn×n λ(A) denotes the n-tuple of eigenvalues of A, s(A) denotes the n-tuple of singular values of A, diag A denotes the diagonal of A. Theorem 1.1. (Schur-Horn) Let λ, d ∈ n. Then there exists a Hermitian R Home Page A ∈ Cn×n such that λ(A) = λ and diag A = d if and only if Title Page k k X X Contents dj ≤ λj, k = 1, . , n − 1, JJ II j=1 j=1 n n X X J I dj = λj Page 2 of 30 j=1 j=1 Go Back after rearranging the entries of d and λ in descending order, respectively. It Full Screen is equivalent to Close d ∈ convSnλ, Quit where “conv” denotes the convex hull of the underlying set, and Snλ denotes the orbit of λ under the action of the symmetric group Sn. The concept is known as majorization, denoted by d ≺ λ. • I. Schur, Uber¨ eine Klasse von Mittelbildungen mit Anwendungen auf der Determinantentheorie, Sitzungsberichte der Berlinear Mathematischen Gesellschaft, 22 (1923), 9–20. • A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math. 76 (1954), 620–630. Rewriting Schur-Horn’s result in orbit terms: Home Page −1 diag {Udiag (λ1, . , λn)U : U ∈ U(n)} = convSnλ. Title Page Mirsky asked the analog for the singular values and diagonal entries of Contents A ∈ Cn×n. JJ II J I • L. Mirsky, Inequalities and existence theorems in the theory of matrices, J. Math. Anal. Appl. 9 Page 3 of 30 (1964), 99–118. Partial results were obtained. Go Back Full Screen • G. de Oliveira, Matrices with prescribed principal elements and singular values, Canad. Math. Bull. Close 14 (1971), 247–249. Quit • F.Y. Sing, Some results on matrices with prescribed diagonal elements and singular values, Canad. Math. Bulletin, 19 (1976), 89–92. n n Theorem 1.2. (Thompson-Sing) Let s ∈ R+ and let d ∈ C . Then there exists A ∈ Cn×n with s(A) = s and diag A = d if and only if k k X X |di| ≤ si, k = 1, . , n, i=1 i=1 Home Page n−1 n−1 X X Title Page |di| − |dn| ≤ si − sn, Contents i=1 i=1 JJ II after rearranging the entries of s and d in descending order with respect to modulus. J I Page 4 of 30 • F.Y. Sing, Some results on matrices with prescribed diagonal elements and singular values, Canad. Go Back Math. Bulletin, 19 (1976), 89–92. Full Screen • R.C. Thompson, Singular values, diagonal elements and convexity, SIAM J. Appl. Math. 32 (1977), Close 39–63. Quit Citations From References: 21 Previous Up Next Article From Reviews: 3 MR0424847 (54 #12805) 15A18 (65F15) Thompson, R. C. Singular values, diagonal elements, and convexity. SIAM J. Appl. Math. 32 (1977), no. 1, 39–63. Rather more than twenty years ago, A. Horn [Amer. J. Math. 76 (1954), 620–630; MR0063336 (16,105c)] found, inter alia, necessary and sufficient conditions for the existence of a Hermi- tian matrix with prescribed diagonal elements and characteristic roots. (An alternative treatment was given subsequently by the reviewer [J. London Math. Soc. 33 (1958), 14–21; MR0091931 (19,1034c)].) Horn’s initiative (in the paper cited above and in a few other notes) became the start- ing point of a distinctive tendency within matrix theory, and the work reviewed here belongs to the tradition so established. Probably the most arresting conclusion reached by the author, which Home Page settles a difficult problem of long standing, is the following: Let d1, ··· , dn be complex numbers arranged so that |d1| ≥ · · · ≥ |dn|, and let s1 ≥ · · · ≥ sn be non-negative real numbers; then there Title Page exists a (complex) n × n matrix with diagonal elements d1, ··· , dn (in any prescribed order) and Pk Pk Pn−1 singular values s1, ··· , sn if and only if i=1 |di| ≤ i=1 si (1 ≤ k ≤ n) and i=1 |di| − |dn| ≤ Contents Pn−1 i=1 si − sn. These inequalities have a familiar look. Nevertheless, the proof is neither easy nor very short and exhibits a high degree of technical ingenuity. The author goes on to derive a variant JJ II of the theorem just stated for the case when all numbers are real and the matrix is also required to have a non-negative determinant. In the final two sections of the paper, questions of convexity are considered; in this field, too, Horn J I [op. cit.] had made a start. The discussion is mainly concerned with a variety of characterizations Page 5 of 30 involving singular values; it is again very solid and detailed and is not readily summarized in a few sentences. We content ourselves with recalling a result of Horn’s which here emerges as a trivial Go Back corollary: The real numbers d1, ··· , dn are the diagonal elements of a proper orthogonal matrix if and only if (d1, ··· , dn) lies in the convex hull of those vectors (±1, ··· , ±1) which have an Full Screen even number of negative components [cf. the reviewer, Amer. Math. Monthly 66 (1959), 19–22; MR0098758 (20 #5213)]. In the reviewer’s opinion, the paper discussed here represents an advance of almost the same Close order of magnitude as the earlier work of Horn’s. In particular, our understanding of convexity properties of matrices is, as yet, very imperfect and it seems likely that the author’s results will Quit help us to gain a more complete insight. Anyone interested in matrix theory would do well to study the paper, to develop further the techniques introduced here, and (if possible) to simplify the rather intricate arguments. Reviewed by L. Mirsky c Copyright American Mathematical Society 1977, 2011 424847.pdf 424850.pdf Citations From References: 10 Previous Up Next Article From Reviews: 0 MR0424850 (54 #12808) 15A42 Sing, Fuk Yum Some results on matrices with prescribed diagonal elements and singular values. Canad. Math. Bull. 19 (1976), no. 1, 89–92. The author considers the relationship between the diagonal elements and singular values of a matrix. This problem originated with L. Mirsky in a well-known survey paper [J. Math. Anal. Appl. 9 (1964), 99–118; MR0163918 (29 #1217)] and was revived by G. N. de Oliveira [Canad. Math. Bull. 14 (1971), 247–249; MR0311686 (47 #248)] who gave a rather simple inequality relating the diagonal elements and singular values. In the paper under review the author obtains Home Page a different condition which implies de Oliveira’s condition, and he also completely discusses the Title Page case of 2 × 2 matrices. In a note added in proof, he observes that he later proved his necessary condition to be sufficient for the existence of an n × n matrix with prescribed diagonal elements Contents and singular values, but that this fact had already been proved by the reviewer. {Reviewer’s remark: The discovery of the necessary and sufficient conditions for the existence JJ II of a matrix with prescribed diagonal elements and singular values had been announced by the reviewer in an abstract in the Notices of the American Math. Society; unfortunately, this an- J I nouncement was overlooked by the author. The reviewer’s paper [SIAM J. Appl. Math. 32 (1977), Page 6 of 30 no. 1, 39–63] (not the SIAM Journal of Mathematical Analysis as stated by the author), contains not only this result but also many related theorems and corollaries. Three further papers of the re- Go Back viewer exploiting these results are to be found [Linear and Multilinear Algebra 3 (1975/76), no. 1/2, 15–17; MR0414581 (54 #2682); ibid. 3 (1975/76), no. 1/2, 155–160; ibid. to appear]. The au- Full Screen thor’s work apparently formed his thesis for his Master’s degree. It is unfortunate that his work Close was anticipated. He plainly is a talented mathematician from whom many more worthwhile results can be expected.} Quit Reviewed by R. C. Thompson c Copyright American Mathematical Society 1977, 2011 n n Theorem 1.3. (Thompson) Let s ∈ R+ and let d ∈ R . Then there exists A ∈ Rn×n with det A ≥ 0 (det A ≤ 0) having s(A) = s and diag A = d if and only if k k X X |di| ≤ si, k = 1, . , n, i=1 i=1 n−1 n−1 Home Page X X |di| − |dn| ≤ si − sn, Title Page i=1 i=1 Contents and in addition, if the number of negative terms among d is odd (even, if JJ II nonpositive determinant) J I n n−1 X X Page 7 of 30 |d | ≤ s − s , i i n Go Back i=1 i=1 Full Screen after rearranging the entries of s and d in descending order with respect to Close absolute value. Quit • The set is a convex set. n n Theorem 1.4. (Thompson) Let s ∈ R+ and let d ∈ R . Then there exists a real matrix with s(A) = s and diag A = d if and only if k k X X |di| ≤ si, k = 1, . , n, (1) Home Page i=1 i=1 n−1 n−1 Title Page X X |di| − |dn| ≤ si − sn, (2) Contents i=1 i=1 JJ II after rearranging d’s in descending order with respect to absolute value.
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