On relationships between two linear subspaces and two orthogonal projectors

References

[1] S. Afriat. On the latent vectors and characteristic values of products of pairs of symmetric idempotents. Quart. J. Math. 7(1956), 76–78.10.1093/qmath/7.1.76Search in Google Scholar

[2] S. Afriat. Orthogonal and oblique projectors and the characteristics of pairs of vector spaces. Proc. Cambridge Phil. Soc. 53(1957), 800–816.10.1017/S0305004100032916Search in Google Scholar

[3] W.O. Amrein, K.B. Sinha. On pairs of projections in a Hilbert space. Linear Algebra Appl. 208/209(1994), 425–435.10.1016/0024-3795(94)90454-5Search in Google Scholar

[4] E. Andruchow, G. Corach. Essentially orthogonal subspaces. J. Oper. Theory 79(2018), 79–100.Search in Google Scholar

[5] A. Arias, S. Gudder. Almost sharp quantum effects. J. Math. Phys. 45(2004), 4196–4206.Search in Google Scholar

[6] J.E. Avron, R. Seiler, B. Simon. The index of a pair of projections. J. Func. Anal. 120(1994), 220–237.Search in Google Scholar

[7] J.K. Baksalary. Algebraic characterizations and statistical implications of the commutativity of orthogonal projectors. In: Proceedings of the Second International Tampere Conference in Statistics (T. Pukkila and S. Puntanen eds.), University of Tampere, Tampere, Finland, 1987, pp. 113–142.Search in Google Scholar

[8] J.K. Baksalary, O.M. Baksalary. Commutativity of projectors. Linear Algebra Appl. 341(2002), 129–142.10.1016/S0024-3795(01)00350-0Search in Google Scholar

[9] J.K. Baksalary, O.M. Baksalary, P. Kik. Generalizations of a property of orthogonal projectors. Linear Algebra Appl. 420(2007), 1–8.Search in Google Scholar

[10] J.K. Baksalary, O.M. Baksalary, T. Szulc. A property of orthogonal projectors. Linear Algebra Appl. 354(2002), 35–39.10.1016/S0024-3795(02)00337-3Search in Google Scholar

[11] J.K. Baksalary, G.P.H. Styan. Around a formula for the rank of a matrix product with some statistical applications. In Graphs, Matrices, and Designs: Festschrift in Honor of N.J. Pullman on his Sixtieth Birthday (R.S. Rees ed.), Marcel Dekker, New York, 1993, pp. 1–18.10.1201/9780203719916-1Search in Google Scholar

[12] O.M. Baksalary, P. Kik. On commutativity of projectors. Linear Algebra Appl. 417(2006), 31–41.10.1016/j.laa.2006.02.019Search in Google Scholar

[13] O.M. Baksalary, G. Trenkler. Characterizations of EP, normal, Hermitian matrices. Linear Multilinear Algebra 56(2008), 299–304.10.1080/03081080600872616Search in Google Scholar

[14] O.M. Baksalary, G. Trenkler. An alternative approach to characterize the commutativity of orthogonal projectors. Discuss. Math. Probab. Stat. 28(2008), 113–137.Search in Google Scholar

[15] O.M. Baksalary, G. Trenkler. Column space equalities for orthogonal projectors. Appl. Math. Comput. 212(2009), 519–529.Search in Google Scholar

[16] O.M. Baksalary, G. Trenkler. Revisitation of the product of two orthogonal projectors. Linear Algebra Appl. 430(2009), 2813–2833.10.1016/j.laa.2009.01.020Search in Google Scholar

[17] O.M. Baksalary, G. Trenkler. Eigenvalues of functions of orthogonal projectors. Linear Algebra Appl. 431(2009), 2172–2186.10.1016/j.laa.2009.07.023Search in Google Scholar

[18] O.M. Baksalary, G. Trenkler. On angles and distances between subspaces. Linear Algebra Appl. 431(2009), 2243–2260.Search in Google Scholar

[19] O.M. Baksalary, G. Trenkler. On a subspace metric based on matrix rank. Linear Algebra Appl. 432(2010), 1475–1491.Search in Google Scholar

[20] O.M. Baksalary, G. Trenkler. Functions of orthogonal projectors involving the Moore–Penrose inverse. Comput. Math. Appl. 59(2010), 764–778.Search in Google Scholar

[21] O.M. Baksalary, G. Trenkler. On the projectors FF^† and F^†F. Appl. Math. Comput. 217(2011), 10213–10223.Search in Google Scholar

[22] O.M. Baksalary, G. Trenkler. Rank formulae from the perspective of orthogonal projectors. Linear Multilinear Algebra 59(2011), 607–625.10.1080/03081081003716483Search in Google Scholar

[23] O.M. Baksalary, G. Trenkler. On disjoint range matrices. Linear Algebra Appl. 435(2011), 1222–1240.10.1016/j.laa.2011.03.005Search in Google Scholar

[24] O.M. Baksalary, G. Trenkler. On the entries of orthogonal projection matrices. In: Combinatorial Matrix Theory and Generalized Inverses of Matrices, R. Bapat, S. Kirkland, K. Prasad, S. Puntanen (eds), Springer, India, 2013, pp. 101–118.10.1007/978-81-322-1053-5_9Search in Google Scholar

[25] H. Behncke. Projections in Hilbert Space. Tahôku Math. J. 22(1970), 181–183.10.2748/tmj/1178242810Search in Google Scholar

[26] H. Behncke. Projections in Hilbert Space II. Tahôku Math. J. 23(1971), 349–352.10.2748/tmj/1178242586Search in Google Scholar

[27] J. Benítez, X. Liu. A short proof of a matrix decomposition with applications. Linear Algebra Appl. 438(2013), 1398–1414.Search in Google Scholar

[28] J. Benítez, V. Rakocěvić. Applications of CS decomposition in linear combinations of two orthogonal projectors. Appl. Math. Comput. 203(2008), 761–769.Search in Google Scholar

[29] J. Benítez, V. Rakocěvić. On the spectrum of linear combinations of two projections in C -algebras. Linear Multilinear Algebra 58(2010), 673–679.Search in Google Scholar

[30] J. Benítez, V. Rakocěvić. Matrices A such that AA^{† −}A^†A are nonsingular. Appl. Math. Comput. 217(2010), 3493–3503.Search in Google Scholar

[31] A. Ben–Israel. On the geometry of subspaces in Euclidean n-spaces. SIAM J. Appl. Math. 15(1967), 1184–1198.10.1137/0115101Search in Google Scholar

[32] A. Ben–Israel, T.N.E. Greville. Generalized Inverses: Theory and Applications. 2nd ed., Springer, New York, 2003.Search in Google Scholar

[33] A.M. Bikchentaev. On representation of elements of a von Neumann algebra in the form of finite sums of products of projections. Siber. Math. J. 46(2005), 24–34.10.1007/s11202-005-0003-4Search in Google Scholar

[34] A.M. Bikchentaev. Commutativity of projectors and trace characterization on von Neumann algebras. I. Russian Math. 53(2009), 68–71.10.3103/S1066369X0912010XSearch in Google Scholar

[35] A.M. Bikchentaev. Commutation of projections and trace characterization on von Neumann algebras. II. Math. Notes 89(2011), 461–471.10.1134/S0001434611030175Search in Google Scholar

[36] A.M. Bikchentaev. Commutation of projections and characterization of traces on von Neumann algebras. III. Int. J. Theor. Phys. 54(2015), 4482–4493.10.1007/s10773-015-2639-6Search in Google Scholar

[37] A. Björck, G. Golub. Numerical methods for computing angles between linear subspaces. Math. Comput. 27(1973), 579–594.Search in Google Scholar

[38] P.E. Bjørstad, J. Mandel. On the spectra of sums of orthogonal projections with applications to parallel computing. BIT 31(1991), 76–88.Search in Google Scholar

[39] E. Boasso. On the Moore–Penrose inverse, EP Banach space operators, and EP Banach algebra elements. J. Math. Anal. Appl. 339(2008), 1003–1014.10.1016/j.jmaa.2007.07.059Search in Google Scholar

[40] E. Boasso. Factorizations of EP Banach space operators and EP Banach algebra elements. J. Math. Anal. Appl. 379(2011), 245–255.10.1016/j.jmaa.2011.01.008Search in Google Scholar

[41] S. Borac. On the algebra generated by two projections. J. Math. Phys. 36(1995), 863–874.10.1063/1.531162Search in Google Scholar

[42] A. Böttcher, I.M. Spitkovsky. Drazin inversion in the von Neumann algebra generated by two orthogonal projections. J. Math. Anal. Appl. 358(2009), 403–409.Search in Google Scholar

[43] A. Böttcher, I.M. Spitkovsky. A gentle guide to the basics of two projections theory. Linear Algebra Appl. 432(2010), 1412–1459.Search in Google Scholar

[44] A. Böttcher, I.M. Spitkovsky. On certain finite-dimensional algebras generated by two idempotents. Linear Algebra Appl. 435(2011), 1823–1836.Search in Google Scholar

[45] A. Böttcher, I.M. Spitkovsky. Group inversion in certain finite-dimensional algebras generated by two idempotents. Indaga. Math. 23(2012), 715–732.Search in Google Scholar

[46] A. Böttcher, I.M. Spitkovsky.Classification of the finite-dimensional algebras generated by two tightly coupled idempotents. Linear Algebra Appl. 439(2013), 538–551.Search in Google Scholar

[47] S. Brenner. Endomorphism algebras of vector spaces with distinguished sets of subspaces. J. Algebra 6(1967), 100–114.10.1016/0021-8693(67)90016-6Search in Google Scholar

[48] S.L. Campbell, C.D. Meyer. Generalized Inverses of Linear Transformations. Corrected reprint of the 1979 original, Dover, New York, 1991.Search in Google Scholar

[49] S. Cheng, Y. Tian. Moore–Penrose inverses of products and differences of orthogonal projectors. Acta Sci. Math. (Szeged) 69(2003), 533–542.Search in Google Scholar

[50] M.-D. Choi, P.Y. Wu. Convex combinations of projections. Linear Algebra Appl. 136(1990), 25–42.10.1016/0024-3795(90)90019-9Search in Google Scholar

[51] M.-D. Choi, P.Y. Wu. Sums of orthogonal projections. J. Funct. Anal. 267(2014), 384–404.Search in Google Scholar

[52] G. Corach, A. Maestripieri. Products of orthogonal projections and polar decompositions. Linear Algebra Appl. 434(2011), 1594–1609.Search in Google Scholar

[53] J. Dauxois, G.M. Nkiet. Canonical analysis of two Euclidean subspaces and its applications. Linear Algebra Appl. 264(1997), 355–388.Search in Google Scholar

[54] C. Davis. Separation of two linear subspaces. Acta Sci. Math. (Szeged) 19(1958), 172–187.Search in Google Scholar

[55] E.R. De Arellano, N.L. Vasilevsky. Algebras of singular integeral operators generated by three othogonal projections. Integr. Equ. Oper. Theory 25(1996), 277–288.Search in Google Scholar

[56] F. Deutsch. The angle between subspaces of a Hilbert space. In: Approximation Theory, Wavelets and Applications, S. Singh (ed.), Kluwer, Dordrecht, 1995, pp. 107–130.10.1007/978-94-015-8577-4_7Search in Google Scholar

[57] J. Dixmier. Position relative de deux variétés linéaires fermées dans un espace de Hilbert. Rev. Sci. 86(1948), 387–399.Search in Google Scholar

[58] D.S. Djordjević. Product of EP operators on Hilbert spaces. Proc. Amer. Math. Soc. 129(2000), 1727–1731.10.1090/S0002-9939-00-05701-4Search in Google Scholar

[59] D.S. Djordjević. Characterizations of normal, hyponormal and EP operators. J. Math. Anal. Appl. 329(2007), 1181–1190.10.1016/j.jmaa.2006.07.008Search in Google Scholar

[60] D.S. Djordjević, J.J. Koliha. Characterizing hermitian, normal and EP operators. Filomat 21(2007), 39–54.Search in Google Scholar

[61] D. Djordjević, J.J. Koliha, I. Straškraba. Factorization of EP elements in C -algebras. Linear Multilinear Algebra 57(2009), 587–594.10.1080/03081080802264372Search in Google Scholar

[62] D. Drivaliaris, S. Karanasios, D. Pappas. Factorizations of EP operators. Linear Algebra Appl. 429(2008), 1555–1567.10.1016/j.laa.2008.04.026Search in Google Scholar

[63] Y.-N. Dou, Y.-Q. Wang, M.-M. Cui. H.-K. Du. Spectra of anticommutator for two orthogonal projections. Linear Multilinear Algebra 67(2019), 2077–2081.10.1080/03081087.2018.1481358Search in Google Scholar

[64] H.-K. Du, C.-Y. Deng. Moore–Penrose inverses of products and differences of orthogonal projections. Acta Anal. Funct. Appl. 8(2006), 104–109.Search in Google Scholar

[65] H.-K. Du, G. Gao, Y. Wang. Geometry characterizations between two subspaces (in Chinese). Acta Math. Scientia 32(2012), 879–891.Search in Google Scholar

[66] M. Enomoto, Y. Watatani. Relative position of four subspaces in a Hilbert space. Adv. Math. 201(2006), 263–317.Search in Google Scholar

[67] J. Farokhi-Ostad, A.R. Janfada. Products of EP operators on Hilbert C^*-modules. Sahand Commun. Math. Anal. 10(2018), 61–71.Search in Google Scholar

[68] I.S. Feshchenko. On closeness of the sum of n subspaces of a Hilbert space. Ukrainian Math. J. 63(2012), 1566–1622.10.1007/s11253-012-0601-9Search in Google Scholar

[69] P.A. Fillmore. On sums of projections. J. Funct. Anal. 4(1969), 146–152.10.1016/0022-1236(69)90027-5Search in Google Scholar

[70] I.M. Gelfand, V.A. Ponomarev. Problems of linear algebra and classification of quadruples of subspaces in a finite-dimensional vector space. Coll. Math. Spc. Bolyai 5(1970), 163–237.Search in Google Scholar

[71] A. Galántai. Subspaces, angles and pairs of orthogonal projections. Linear Multilinear Algebra 56(2008), 227–260.10.1080/03081080600743338Search in Google Scholar

[72] A. Galántai, Cs.J. Heged¶s. Jordan’s principal angles in complex vector spaces. Numer. Linear Algebra Appl. 13(2006), 589–598.Search in Google Scholar

[73] R. Giles, H. Kummer. A matrix representation of a pair of projections in a Hilbert space. Canad. Math. Bull. 14(1971), 35–44.Search in Google Scholar

[74] G.H. Golub, C.F. Van Loan. Matrix Computations. 3rd edition, Baltimore, Johns Hopkins University Press, 1996.Search in Google Scholar

[75] T.N.E. Greville. Solutions of the matrix equations XAX = X and relations between oblique and orthogonal projectors. SIAM J. Appl. Math. 26(1974), 828–832.10.1137/0126074Search in Google Scholar

[76] J. Groß. On the product of orthogonal projectors. Linear Algebra Appl. 289(1999), 141–150.10.1016/S0024-3795(98)10064-2Search in Google Scholar

[77] J. Groß. On oblique projection, rank additivity and the Moore–Penrose inverse of the sum of two matrices. Linear Multilinear Algebra 46(1999), 265–275.Search in Google Scholar

[78] S. Gudder. Sharp and unsharp quantum effects. Adv. Appl. Math. 20(1998), 169–187.10.1006/aama.1997.0575Search in Google Scholar

[79] P.R. Halmos. Two subspaces. Trans. Amer. Math. Soc. 144(1969), 381–389.10.1090/S0002-9947-1969-0251519-5Search in Google Scholar

[80] P.R. Halmos. Finite-dimensional Hilbert spaces. Amer. Math. Month. 77(1970), 457–464.10.1080/00029890.1970.11992521Search in Google Scholar

[81] P.R. Halmos. Finite-dimensional Vector Spaces. Springer, New York, 1974.10.1007/978-1-4612-6387-6Search in Google Scholar

[82] P.R. Halmos. A Hilbert Space Problem Book. 2nd Ed., Springer, New York 1982.10.1007/978-1-4684-9330-6Search in Google Scholar

[83] I. Halperin. The product of projection operators. Acta Sci. Math. (Szeged) 23(1962), 96–99.Search in Google Scholar

[84] R.E. Hartwig, I.J. Katz. On products of EP matrices. Linear Algebra Appl. 252(1997), 339–345.10.1016/0024-3795(95)00693-1Search in Google Scholar

[85] R.E. Hartwig, K. Spindelböck. Matrices for which A and A^† can commute. Linear Multilinear Algebra 14(1984), 241–256.10.1080/03081088308817561Search in Google Scholar

[86] M. Hegland, J. Garcke, V. Challis. The combination technique and some generalisations. Linear Algebra Appl. 420(2007), 249–275.Search in Google Scholar

[87] Kh.D. Ikramov. A canonical form for projectors under unitary similarity. Comput. Math. Math. Phys. 36(1996), 279–281.Search in Google Scholar

[88] M. Itoh. On some EP operators. Nihonkai Math. J. 16(2005), 49–56.Search in Google Scholar

[89] B. Jiang, Y. Tian. Necessary and sufficient conditions for nonlinear matrix identities to always hold. Aequat. Math. 93(2019), 587–600.Search in Google Scholar

[90] J.J. Koliha. A simple proof of the product theorem for EP matrices. Linear Algebra Appl. 294(1999), 213–215.10.1016/S0024-3795(99)00066-XSearch in Google Scholar

[91] J.J. Koliha. Elements of C^*-algebras commuting with their Moore–Penrose inverse. Studia Math. 139(2000), 81–90.10.4064/sm-139-1-81-90Search in Google Scholar

[92] J.J. Koliha, V. Rakocěvić. Fredholm properties of the difference of orthogonal projections in a Hilbert space. Integr. Equ. Oper. Theory 52(2005), 125–134.Search in Google Scholar

[93] T. Kollo, D. von Rosen. Advanced Multivariate Statistics with Matrices. Springer, 2005.10.1007/1-4020-3419-9Search in Google Scholar

[94] S.A. Kruglyak, V.I. Rabanovich, Yu.S. Samoĭilenko. On sums of projections. Funct. Anal. Appl. 36(2002), 182–195.10.1023/A:1020193804109Search in Google Scholar

[95] S.A. Kruglyak, V.I. Rabanovich, Yu.S. Samoĭilenko. Decomposition of a scalar matrix into a sum of orthogonal projections. Linear Algebra Appl. 370(2003), 217–225.Search in Google Scholar

[96] S.A. Kruglyak, Yu.S. Samoĭilenko. Structural theorems for families of idempotents. Ukrain. Math. J. 50(1998), 593–604.Search in Google Scholar

[97] A.A. Kyrychenko, Yu.S. Samoĭilenko. On the spectrum and spectrum multiplicities of a sum of orthogonal projections. Algebra Disc. Math. 3(2004), 71–76.Search in Google Scholar

[98] A. Lenard. The numerical range of a pair of projections. J. Funct. Anal. 10(1972), 410–423.10.1016/0022-1236(72)90037-7Search in Google Scholar

[99] Y. Li. The Moore–Penrose inverses of products and differences of projections in a C^*-algebra. Linear Algebra Appl. 428(2008), 1169–1177.10.1016/j.laa.2007.09.021Search in Google Scholar

[100] W.E. Longstaff, O. Panaia. On complementary subspaces of Hilbert space. Proc. Amer. Math. Soc. 126(1988), 3019–3026.10.1090/S0002-9939-98-04547-XSearch in Google Scholar

[101] W. López. A new way of computing the orthogonal projection onto the intersection of two hyperplanes in a finite-dimensional Hilbert space. Appl. Math. E-Notes 18(2018), 116–123.Search in Google Scholar

[102] L.W. Marcoux. Projections, commutators and Lie ideals in C^*-algebras. Math. Proc. R. Ir. Acad. 110A(2010), 31–55.10.1353/mpr.2010.0014Search in Google Scholar

[103] G. Marsaglia, G.P.H. Styan. Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2(1974), 269–292.10.1080/03081087408817070Search in Google Scholar

[104] N. Matzakos, D. Pappas. EP matrices: computation of the Moore–Penrose inverse via factorizations. J. Appl. Math. Comput. 34(2010), 113–127.Search in Google Scholar

[105] C.D. Meyer. Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia, 2000.10.1137/1.9780898719512Search in Google Scholar

[106] J. Miao, A. Ben-Israel. On principal angles between subspaces in 𝕉ⁿ. Linear Algebra Appl. 171(1992), 81–98.Search in Google Scholar

[107] J. Miao, A. Ben-Israel. Product cosines of angles between subspaces. Linear Algebra Appl. 237/238(1996), 71–81.10.1016/0024-3795(95)00425-4Search in Google Scholar

[108] Yu.P. Moskaleva, Yu.S. Samoilenko. Systems of n subspaces and representations of -algebras generated by projections. Methods Funct. Anal. Topology 12(2006), 57–73.Search in Google Scholar

[109] H. Nakano. Spectral Theory in the Hilbert Space. Japan Society for the Promotion of Science, Tokyo, 1953.Search in Google Scholar

[110] M. Nees. Products of orthogonal projections as Carleman operators. Integr. Equ. Oper. Theory 35(1999), 85–92.10.1007/BF01225529Search in Google Scholar

[111] K. Nishio. The structure of a real linear combination of two projections. Linear Algebra Appl. 66(1985), 169–176.10.1016/0024-3795(85)90130-2Search in Google Scholar

[112] K. Nordström, D. von Rosen. Algebra of subspaces with applications to problems in statistics. In: Proceedings of the Second International Tampere Conference in Statistics (T. Pukkila and S. Puntanen eds.), University of Tampere, Tampere, Finland, 1987, pp. 603–614.Search in Google Scholar

[113] T. Oikhberg. Products of orthogonal Projections. Proc. Amer. Math. Soc. 127(1999), 3659–3669.10.1090/S0002-9939-99-05255-7Search in Google Scholar

[114] C.C. Paige, M. Wei. History and generality of the CS decomposition. Linear Algebra Appl. 1994, 208/209(1994), 303–326.Search in Google Scholar

[115] P.V. Pethe, N.K. Thakare. Convex sums of orthogonal projections on a Banach space. Math. Student 45(1977), 49–52.Search in Google Scholar

[116] R. Piziak, P.L. Odell, R. Hahn. Constructing projections on sums and intersections. Comput. Math. Appl. 37(1999), 67–74.Search in Google Scholar

[117] S. Puntanen, G.P.H. Styan, J. Isotalo. Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty. Springer, 2011.10.1007/978-3-642-10473-2Search in Google Scholar

[118] I. Raeburn, A.M. Sinclair. The C^*-algebra generated by projections. Math. Scandinavica 65(1989), 278–290.Search in Google Scholar

[119] V. Rakocěvić, H.K. Wimmer. A variational characterization of canonical angles between subspaces. J. Geometry 78(2003), 122–124.Search in Google Scholar

[120] C.R. Rao, S.K. Mitra. Generalized Inverse of Matrices and Its Applications. Wiley, New York, 1971.Search in Google Scholar

[121] W. Rehder. Spectral properties of products of projections in quantum probability theory. Internat. J. Theor. Phys. 18(1979), 791–805.10.1007/BF00670458Search in Google Scholar

[122] W. Rehder. On the commutativity of two projections. Elem. Math. 35(1980), 120–122.10.1515/zna-1980-0415Search in Google Scholar

[123] W. Rehder. When do projections commute? Z. Naturforsch. 35A(1980), 437–441.10.1515/zna-1980-0415Search in Google Scholar

[124] I.B. Risteski, K.G. Trencevski. Principal values and principal subspaces of two subspaces of vector spaces with inner product. Beitr. Algebra Geom. 42(2001), 289–300.Search in Google Scholar

[125] A.G. Robertson. Linear and convex combinations of projections in simple C^*-algebras. Proc. Roy. Soc. Edinburgh Sect. A 101(1985), 125–129.10.1017/S0308210500026202Search in Google Scholar

[126] S. Roman. Advanced Linear Algebra. Springer, New York, 1992.10.1007/978-1-4757-2178-2Search in Google Scholar

[127] H. Schwerdtfeger. Introduction to Linear Algebra and the Theory of Matrices. Noordhoff, Groningen, 1950.Search in Google Scholar

[128] K. Sharifi. EP modular operators and their products. J. Math. Anal. Appl. 419(2014), 870–877.10.1016/j.jmaa.2014.05.031Search in Google Scholar

[129] W. Shi, G. Ji, H.-K. Du. Pairs of orthogonal projections with a fixed difference. Linear Algebra Appl. 489(2016), 288–297.Search in Google Scholar

[130] G.W. Stewart. Computing the CS decomposition of a partitioned orthonormal matrix. Numer. Math. 40(1982), 297–306.10.1007/BF01396447Search in Google Scholar

[131] I.M. Spitkovsky. Once more on algebras generated by two projections. Linear Algebra Appl. 208/209(1994), 377–395.10.1016/0024-3795(94)90450-2Search in Google Scholar

[132] I.M. Spitkovsky. On polynomials in two projections. Electron. J. Linear Algebra 15(2006), 154–158.10.13001/1081-3810.1226Search in Google Scholar

[133] Y. Tian. Upper and lower bounds for ranks of matrix expressions using generalized inverses. Linear Algebra Appl. 355(2002), 187–214.10.1016/S0024-3795(02)00345-2Search in Google Scholar

[134] Y. Tian. The dimension of the intersection of k subspaces. Missouri J. Math. Sci. 14(2002), 1–4.10.35834/2002/1402092Search in Google Scholar

[135] Y. Tian. Using rank formulas to characterize equalities for Moore–Penrose inverses of matrix products. Appl. Math. Comput. 147(2004), 581–600.10.1016/S0096-3003(02)00796-8Search in Google Scholar

[136] Y. Tian. Equalities for orthogonal projectors and their operations. Centr. Eur. J. Math. 8(2010), 855–870.10.2478/s11533-010-0057-9Search in Google Scholar

[137] Y. Tian. Eight expressions for generalized inverses of a bordered matrix. Linear Multilinear Algebra 58(2010), 203–220.10.1080/03081080802095461Search in Google Scholar

[138] Y. Tian. Expansion formulas for the inertias of Hermitian matrix polynomials and matrix pencils of orthogonal projectors. J. Math. Anal. Appl. 376(2011), 162–186.10.1016/j.jmaa.2010.09.038Search in Google Scholar

[139] Y. Tian. Formulas for calculating the dimensions of the sums and the intersections of a family of linear subspaces with applications. Beitr. Algebra Geom. 60(2019), 471–485.10.1007/s13366-018-00432-9Search in Google Scholar

[140] Y. Tian. Linear statistical models, least-squares estimators, and classification analysis to reverse-order laws for generalized inverses of matrix products. doi:10.20944/preprints201812.0342.v1.10.20944/preprints201812.0342.v1Search in Google Scholar

[141] Y. Tian, Y. Li. Distributions of eigenvalues and inertias of some block Hermitian matrices consisting of orthogonal projectors.Search in Google Scholar

[142] Y. Tian, G.P.H. Styan. Rank equalities for idempotent and involutory matrices. Linear Algebra Appl. 335(2001), 101–117.Search in Google Scholar

[143] Y. Tian, G.P.H. Styan. Rank equalities for idempotent matrices with applications. J. Comput. Appl. Math. 191(2006), 77–97.Search in Google Scholar

[144] Y. Tian, H. Wang. Characterizations of EP matrices and weighted-EP matrices. Linear Algebra Appl. 34(2011), 1295–1318.Search in Google Scholar

[145] Y. Tian, Y. Wang. Expansion formulas for orthogonal projectors onto ranges of row block matrices. J. Math. Res. Appl. 34(2014), 147–154.Search in Google Scholar

[146] G. Trenkler. Characterizations of oblique and orthogonal projectors. In: Proceedings of the International Conference on Linear Statistical Inference LINSTAT ’93, T. Cali/’nski, R. Kala (eds.), Mathematics and Its Applications, vol 306. Springer, Dordrecht 1994, pp. 255–270.10.1007/978-94-011-1004-4_28Search in Google Scholar

[147] G. Trenkler. Problem 31-7: On the product of orthogonal projectors. IMAGE, Bull. Internat. Linear Algebra Soc. 31(2003), 43.Search in Google Scholar

[148] N.L. Vasilevsky. C^*-algebras generated by othogonal projections and their applications. Integr. Equ. Oper. Theory 31(1998), 113–132.10.1007/BF01203459Search in Google Scholar

[149] N.L. Vasilevsky, I.M. Spitkovsky. On the algebra generated by two projections (in Russian). Dokl. Akad. Nauk Ukrain. SSR Ser. A 8(1981), 10–13.Search in Google Scholar

[150] P. Wedin. On angles between subspaces of a finite dimensional inner product space. In: Matrix Pencils, Lecture Notes Math. 973(1983), 263–285.10.1007/BFb0062107Search in Google Scholar

[151] H.K. Wimmer. Canonical angles of unitary spaces and perturbations of direct complements. Linear Algebra Appl. 287(1999), 373–379.10.1016/S0024-3795(98)10017-4Search in Google Scholar

[152] N. Zhang, Y. Wei. Solving EP singular linear systems. Internat. J. Comput. Math. 81(2004), 1395–1405.Search in Google Scholar

[153] X. Zhang, S. Zhang, J. Chen, L. Wang. Moore–Penrose invertibility of differences and products of projections in rings with involution. Linear Algebra Appl. 439(2013), 4101–4109.Search in Google Scholar

[154] K. Zuo, T. Xie. Nonsingularity of the combinations of two orthogonal projectors. Wuhan Univ. J. Nat. Sci. 17(2012), 7–11.10.1007/s11859-012-0796-5Search in Google Scholar