Decomposition of a Hermitian matrix into a sum of a fixed number of orthoprojections

Keywords: Orthoprojection, Hermitian matrix, Horn inequlities, Frame.

Abstract

We prove that any Hermitian matrix, whose trace is integer and all eigenvalues lie in $[1+1/(k-3),k-1-1/(k-3)],$ is a sum of $k$ orthoprojections. For sums of $k$ orthoprojections, it is shown that the ratio of the number of eigenvalues not exceeding 1 to the number of eigenvalues not less than 1, taking into account the multiplicity, is not greater than $k-1$. Examples of Hermitian matrices that satisfy the ratio for eigenvalues and, at the same time, can not be decomposed into a sum of $k$ orthoprojections are also suggested.

References

Finite frames. Theory and applications. Edited by Peter G. Casazza and Gitta Kutyniok. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York, 2013. xvi+483 pp. ISBN: 978-0-8176-8372-6; 978-0-8176-8373-3 https://doi.org/10.1007/978-0-8176-8373-3_13 DOI: https://doi.org/10.1007/978-0-8176-8373-3_13

Calderbank, Robert; Casazza, Peter G.; Heinecke, Andreas; Kutyniok, Gitta; Pezeshki, Ali. Sparse fusion frames: existence and construction. Adv. Comput. Math. 35 (2011), no. 1, 1–31. https://doi.org/10.1007/s10444-010-9162-3 DOI: https://doi.org/10.1007/s10444-010-9162-3

Casazza, Peter G.; Fickus, Matthew; Mixon, Dustin G.; Wang, Yang; Zhou, Zhengfang. Constructing tight fusion frames. Appl. Comput. Harmon. Anal. 30 (2011), no. 2, 175–187. https://doi.org/10.1016/j.acha.2010.05.002 DOI: https://doi.org/10.1016/j.acha.2010.05.002

Leng, Jinsong; Han, Deguang. Orthogonal projection decomposition of matrices and construction of fusion frames. Adv. Comput. Math. 38 (2013), no. 2, 369–381. https://doi.org/10.1007/s10444-011-9241-0 DOI: https://doi.org/10.1007/s10444-011-9241-0

Bjørstad, Petter E.; Mandel, Jan. On the spectra of sums of orthogonal projections with applications to parallel computing. BIT 31 (1991), no. 1, 76–88. https://doi.org/10.1007/bf01952785 DOI: https://doi.org/10.1007/BF01952785

Nishio, Katsuyoshi. The structure of a real linear combination of two projections. Linear Algebra Appl. 66 (1985), 169–176. https://doi.org/10.1016/0024-3795(85)90130-2 DOI: https://doi.org/10.1016/0024-3795(85)90130-2

Ostrovs'kyj, V. L.;, Jakymenko, D. Ju. Про iснування та побудову ортоскалярних наборiв пiдпросторiв. (Ukrainian) [Pro isnuvannja ta pobudovu ortoskaljarnyh naboriv pidprostoriv]. Зб. праць Iн-ту математики НАН України [Zb. prac' In-tu matematyky NAN Ukrai'ny], 12, no. 1, 154–165 (2015). http://www.irbis-nbuv.gov.ua/cgi-bin/irbis_nbuv/cgiirbis_64.exe?I21DBN=LINK&P21DBN=UJRN&Z21ID=&S21REF=10&S21CNR=20&S21STN=1&S21FMT=ASP_meta&C21COM=S&2_S21P03=FILA=&2_S21STR=Zpim_2015_12_1_10

Böttcher, A.; Spitkovsky, I. M. A gentle guide to the basics of two projections theory. Linear Algebra Appl. 432 (2010), no. 6, 1412–1459. https://doi.org/10.1016/j.laa.2009.11.002 DOI: https://doi.org/10.1016/j.laa.2009.11.002

Fillmore, Peter A. On sums of projections. J. Functional Analysis 4 1969 146–152. https://doi.org/10.1016/0022-1236(69)90027-5 DOI: https://doi.org/10.1016/0022-1236(69)90027-5

Kruglyak, Stanislav; Rabanovich, Vyacheslav; Samoĭlenko, Yuriĭ. Decomposition of a scalar matrix into a sum of orthogonal projections. Linear Algebra Appl. 370 (2003), 217–225. https://doi.org/10.1016/s0024-3795(03)00390-2 DOI: https://doi.org/10.1016/S0024-3795(03)00390-2

Fulton, William. Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209–249. https://doi.org/10.1090/s0273-0979-00-00865-x DOI: https://doi.org/10.1090/S0273-0979-00-00865-X

Fulton, William. Eigenvalues of majorized Hermitian matrices and Littlewood–Richardson coefficients. Special Issue: Workshop on Geometric and Combinatorial Methods in the Hermitian Sum Spectral Problem (Coimbra, 1999). Linear Algebra Appl. 319 (2000), no. 1-3, 23–36. https://doi.org/10.1016/s0024-3795(00)00218-4 DOI: https://doi.org/10.1016/S0024-3795(00)00218-4

Horn, Roger A.; Johnson, Charles R. Matrix analysis. Second edition. Cambridge University Press, Cambridge, 2013. xviii+643 pp. ISBN: 978-0-521-54823-6 https://www.researchgate.net/deref/http%3A%2F%2Fdx.doi.org%2F10.1016%2Fj.laa.2014.10.023

Kruglyak, S. A.; Rabanovich, V. I.; Samoĭlenko, Yu. S. On sums of projections. (Russian); translated from Funktsional. Anal. i Prilozhen. 36 (2002), no. 3, 20–35, Funct. Anal. Appl. 36 (2002), no. 3, 182–195 https://doi.org/10.1023/a:1020193804109 DOI: https://doi.org/10.1023/A:1020193804109

Wang, Jin Hsien. The length problem for a sum of idempotents. Linear Algebra Appl. 215 (1995), 135–159. https://doi.org/10.1016/0024-3795(93)00083-c DOI: https://doi.org/10.1016/0024-3795(93)00083-C

Wu, Pei Yuan. Additive combinations of special operators. Functional analysis and operator theory (Warsaw, 1992), 337–361, Banach Center Publ., 30, Polish Acad. Sci. Inst. Math., Warsaw, 1994. https://doi.org/10.4064/-30-1-337-361 DOI: https://doi.org/10.4064/-30-1-337-361

Published
29.04.2020
How to Cite
RabanovichV. I. “Decomposition of a Hermitian Matrix into a Sum of a Fixed Number of Orthoprojections”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 5, Apr. 2020, pp. 679–693, doi:10.37863/umzh.v72i5.2378.
Section
Research articles