An Analysis of Equality in Certain Matrix Inequalities. II
| Autor: | Marvin Marcus, William R. Gordon |
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| Rok vydání: | 1972 |
| Předmět: | |
| Zdroj: | SIAM Journal on Numerical Analysis. 9:130-136 |
| ISSN: | 1095-7170 0036-1429 |
| DOI: | 10.1137/0709013 |
| Popis: | In this paper the authors investigate the cases of equality in some generalizations of Schur’s inequality relating the eigenvalues and singular values of an $n \times n$ complex matrix. For example, it follows from the results in the paper that if A is an $n \times n$ complex matrix with eigenvalues $\lambda _1 , \cdots ,\lambda _n $ (with $|\lambda _1 | \geqq \cdots \geqq |\lambda _n $) and singular values $\alpha _1 \geqq \cdots \geqq \alpha _n $ and if $h(x_1 , \cdots ,x_k )$ is a nonzero symmetric polynomial in $x_1 , \cdots ,x_k ,1 \leqq k \leqq n$, then for $q > 0$, \[h(|\lambda _1 |^q , \cdots ,|\lambda _k |^q ) \leqq h(\alpha _1^q , \cdots ,\alpha _k^q );\] if $h(x_1 , \cdots ,x_k )$ is not of the form $\Sigma _{j = 0}^r d_j (\Pi _{t = 1}^k x_t )^j ,d_j \geqq 0$, then equality holds only if there is a unitary matrix U such that $U^ * AU = {\operatorname{diag}} (\lambda _1 , \cdots ,\lambda _k ) + T$, where T is an $(n - k)$-square upper triangular matrix. |
| Databáze: | OpenAIRE |
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