Invertibility of linear combinations of two idempotents
HTML articles powered by AMS MathViewer
- by Hongke Du, Xiyan Yao and Chunyuan Deng PDF
- Proc. Amer. Math. Soc. 134 (2006), 1451-1457 Request permission
Abstract:
Let $P$ and $Q$ be two idempotents on a Hilbert space. In this note, we prove that the invertibility of the linear combination $\lambda _1P+\lambda _2Q$ is independent of the choice of $\lambda _i$, $i=1,2,$ if $\lambda _1\lambda _2\neq 0$ and $\lambda _1+\lambda _2\neq 0.$References
- Jerzy K. Baksalary and Oskar Maria Baksalary, Nonsingularity of linear combinations of idempotent matrices, Linear Algebra Appl. 388 (2004), 25–29. MR 2077846, DOI 10.1016/j.laa.2004.02.025
- Oskar Maria Baksalary, Idempotency of linear combinations of three idempotent matrices, two of which are disjoint, Linear Algebra Appl. 388 (2004), 67–78. MR 2077851, DOI 10.1016/S0024-3795(02)00545-1
- Man Duen Choi and Pei Yuan Wu, Convex combinations of projections, Linear Algebra Appl. 136 (1990), 25–42. MR 1061537, DOI 10.1016/0024-3795(90)90019-9
- G. Corach, A. Maestripieri, and D. Stojanoff, Generalized Schur complements and oblique projections, Linear Algebra Appl. 341 (2002), 259–272. Special issue dedicated to Professor T. Ando. MR 1873624, DOI 10.1016/S0024-3795(01)00384-6
- Jürgen Groß and Götz Trenkler, Nonsingularity of the difference of two oblique projectors, SIAM J. Matrix Anal. Appl. 21 (1999), no. 2, 390–395. MR 1718336, DOI 10.1137/S0895479897320277
- Stanislav Kruglyak, Vyacheslav Rabanovich, and Yuriĭ Samoĭlenko, Decomposition of a scalar matrix into a sum of orthogonal projections, Linear Algebra Appl. 370 (2003), 217–225. MR 1994329, DOI 10.1016/S0024-3795(03)00390-2
- J. J. Koliha, V. Rakočević, and I. Straškraba, The difference and sum of projectors, Linear Algebra Appl. 388 (2004), 279–288. MR 2077865, DOI 10.1016/j.laa.2004.03.008
- Eugene Spiegel, Sums of projections, Linear Algebra Appl. 187 (1993), 239–249. MR 1221707, DOI 10.1016/0024-3795(93)90138-E
Additional Information
- Hongke Du
- Affiliation: College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China
- Email: hkdu@snnu.edu.cn
- Xiyan Yao
- Affiliation: College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China
- Email: yaoxiyan63@163.com
- Chunyuan Deng
- Affiliation: College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, People’s Republic of China
- Email: cy-deng@263.net
- Received by editor(s): June 19, 2004
- Received by editor(s) in revised form: December 20, 2004
- Published electronically: October 18, 2005
- Additional Notes: This research was partially supported by the National Natural Science Foundation of China (19771056)
- Communicated by: Joseph A. Ball
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1451-1457
- MSC (2000): Primary 47A05, 47L07
- DOI: https://doi.org/10.1090/S0002-9939-05-08091-3
- MathSciNet review: 2199192