# An Elementary Approach to an Eigenvalue Estimate for Matrices

An Elementary Approach to an Eigenvalue Estimate for Matrices A celebrated result of Johnson, Maurey, König and Retherford from 1977 states that for $$2 \leqslant p < \infty$$ every complex $$n \times n$$ matrix $$T = (\tau _{ij} )_{i,j}$$ satisfies the following eigenvalue estimate: $$\left( {\sum\limits_{i = 1}^n {\left| {\lambda _i \left( T \right)} \right|^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \leqslant \left( {\sum\limits_{j = 1}^n {\left( {\sum\limits_{i = 1}^n {\left| {\tau _{ij} } \right|^{p'} } } \right)} {p \mathord{\left/ {\vphantom {p {p'}}} \right. \kern-\nulldelimiterspace} {p'}}} \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \cdot$$ Based on the concept of entropy numbers and a simple product trick we give a selfcontained elementary proof. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# An Elementary Approach to an Eigenvalue Estimate for Matrices

, Volume 4 (2) – Oct 25, 2004
11 pages

/lp/springer_journal/an-elementary-approach-to-an-eigenvalue-estimate-for-matrices-MLSOaUOnEW
Publisher
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009838325440
Publisher site
See Article on Publisher Site

### Abstract

A celebrated result of Johnson, Maurey, König and Retherford from 1977 states that for $$2 \leqslant p < \infty$$ every complex $$n \times n$$ matrix $$T = (\tau _{ij} )_{i,j}$$ satisfies the following eigenvalue estimate: $$\left( {\sum\limits_{i = 1}^n {\left| {\lambda _i \left( T \right)} \right|^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \leqslant \left( {\sum\limits_{j = 1}^n {\left( {\sum\limits_{i = 1}^n {\left| {\tau _{ij} } \right|^{p'} } } \right)} {p \mathord{\left/ {\vphantom {p {p'}}} \right. \kern-\nulldelimiterspace} {p'}}} \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} \cdot$$ Based on the concept of entropy numbers and a simple product trick we give a selfcontained elementary proof.

### Journal

PositivitySpringer Journals

Published: Oct 25, 2004

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