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Two Nonzero Component Lemma and Matrix Trigonometry
Indiana University East, Richmond, USA
- 1 Indiana University East, Richmond, USA
Advances in Linear Algebra & Matrix Theory·Volume 07 (2017)·Pages 1–6·Published 15 February 2017·DOI10.4236/alamt.2017.71001
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Abstract
In this paper we show that the author’s Two Nonzero Lemma (TNCL) can be applied to present a simple proof for a very useful equality which was first proved by Karl Gustafson in 1968. Gustafson used Hilbert space methods, including convexity of the Hilbert space norm, to prove this identity which was the basis of his matrix trigonometry. By applying TNCL, we will reduce the problem to a simple problem of ordinary calculus.
KeywordsMatrix Trigonometry
- Gustafson, K. (2010) On My Min-Max Theorem (1968) and Its Consequences. Acta et Commentationes Universitatis Tartuensis de Mathematica, 14, 45-51.
- Gustafson, K. (1968) A Min-Max Theorem. Notices of the American Mathematical Society, 15, 799.
- Gustafson, K. and Rao, D. (1997) Numerical Range. Springer. https://doi.org/10.1007/978-1-4613-8498-4
- Gustafson, K. and Seddighin, M. (1989) Antieigenvalue Bounds. Journal of Mathematical Analysis and Applications, 143, 327-340. https://doi.org/10.1016/0022-247X(89)90044-9
- Seddighin, M. (2002) Antieigenvalues and Total Antieigenvalues of Normal Operators. Journal of Mathematical Analysis and Applications, 274, 239-254. https://doi.org/10.1016/S0022-247X(02)00295-0
- Seddighin, M. (2009) Antieigenvalue Techniques in Statistics. Linear Algebra and Its Applications, 430, 2566-2580. https://doi.org/10.1016/j.laa.2008.05.007
- Seddighin, M. and Gustafsib, K. (2005) On the Eigenvalues which Express Antieigenvalues. International Journal of Mathematics and Mathematical Sciences, 2005, 1543-1554.
- Seddighin, M. (2010) Gustafson K. Slant Antieigenvalues and Slant Antieigenvectors of Operators. Journal of Linear Algebra and Applications, 432, 1348-1362. https://doi.org/10.1016/j.laa.2009.11.001
- Seddighin, M. (2011) Slant Joint Antieigenvalues and Antieigenvectors of Operators in Normal Subalgebras. Journal of Linear Algebra and its Applications, 434, 1395-1408. https://doi.org/10.1016/j.laa.2010.11.020
- Seddighin, M. (2014) Proving and Extending Greub-Reinboldt Inequality Using the Two Nonzero Component Lemma. Advances in Linear Algebra & Matrix Theory, 4, 120-127. https://doi.org/10.4236/alamt.2014.42010
- Seddighin, M. (2014) Application of the Two Nonzero Component Lemma in Resource Allocation. Journal of Applied Mathematics and Physics, 2, 653-661.