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Matrices with prescribed Ritz values

✍ Scribed by Beresford Parlett; Gilbert Strang

Publisher
Elsevier Science
Year
2008
Tongue
English
Weight
205 KB
Volume
428
Category
Article
ISSN
0024-3795

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✦ Synopsis

On the way to establishing a commutative analog to the Gelfand-Kirillov theorem in Lie theory, Kostant and Wallach produced a decomposition of M(n) which we will describe in the language of linear algebra. The "Ritz values" of a matrix are the eigenvalues of its leading principal submatrices of order m = 1, 2, . . . , n. There is a unique unit upper Hessenberg matrix H with those eigenvalues. For real symmetric matrices with interlacing Ritz values, we extend their analysis to allow eigenvalues at successive levels to be equal. We also decide whether given Ritz values can come from a tridiagonal matrix.

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In this paper we study singular values of a matrix whose one entry varies while all other entries are prescribed. In particular, we find the possible pth singular value of such a matrix, and we define explicitly the unknown entry such that the completed matrix has the minimal possible pth singular v