Formulas for calculating the dimensions of the sums and the intersections of a family of linear subspaces with applications

Baksalary, J.K., Baksalary, O.M.: An invariance property related to the reverse order law. Linear Algebra Appl. 410, 64–69 (2005)

Baksalary, J.K., Kala, R.: Range invariance of certain matrix products. Linear Multilinear Algebra 14, 89–96 (1983)

Baksalary, J.K., Mathew, T.: Linear sufficiency and completeness in an incorrectly specified general Gauss–Markov model. Sankhyā Ser A. 48, 169–180 (1986)

Baksalary, J.K., Pukkila, T.: A note on invariance of the eigenvalues, singular values, and norms of matrix products involving generalized inverses. Linear Algebra Appl. 165, 125–130 (1992)

Baksalary, O.M., Trenkler, G.: On a subspace metric based on matrix rank. Linear Algebra Appl. 432, 1475–1491 (2010)

Ben-Israel, A.: On the geometry of subspaces in euclidean n-spaces. SIAM J. Appl. Math. 15, 1184–1198 (1967)

Ben-Israel, A., Greville, T.N.E.: Generalized inverses: theory and applications, 2nd edn. Springer, New York (2003)

Campbell, S.L., Meyer, C.D.: Generalized inverses of linear transformations. Corrected reprint of the 1979 original. Dover Publications, New York (1991)

Friberg, H.A.: A relaxed-certificate facial reduction algorithm based on subspace intersection. Operat. Res. Lett. 44, 718–722 (2016)

Groß, J., Tian, Y.: Invariance properties of a triple matrix product involving generalized inverses. Linear Algebra Appl. 417, 94–107 (2006)

Halmos, P.R.: Finite-Dimensional Vector Spaces. Springer, New York (1974)

Jiang, B., Tian, Y.: Necessary and sufficient conditions for nonlinear matrix identities to always hold. Aequat Math. (2018). https://doi.org/10.1007/s00010-018-0610-3

Kollo, T., von Rosen, D.: Advanced Multivariate Statistics With Matrices. Springer, Dordrecht (2005). Kindly check and confirm publisher location for reference (Kollo 2005)

Kornacki, A.: Stability of invariant linearly sufficient statistics in the general Gauss–Markov model. Appl. Math. 42, 71–77 (1997)

Lebedinskaya, N.A., Lebedinskii, D.M.: On the possible dimensions of subspace intersections. Vestnik St. Petersburg Univ. Math. 49, 115–118 (2016)

Lebedinskaya, N.A., Lebedinskii, D.M., Smirnov, A.A.: On possible dimensions of subspace intersections for five direct summands. J. Math. Sci. 224, 937–941 (2017)

López, W.: 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, 116–123 (2018)

Marsaglia, G., Styan, G.P.H.: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2, 269–292 (1974)

Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000)

Piziak, R., Odell, P.L., Hahn, R.: Constructing projections on sums and intersections. Comput. Math. Appl. 37, 67–74 (1999)

Puntanen, S., Styan, G.P.H., Isotalo, J.: Matrix Tricks for Linear Statistical Models: Our Personal Top Twenty. Springer, Berlin (2011)

Rao, C.R., Mitra, S.K.: Generalized Inverse of Matrices and Its Applications. Wiley, New York (1971)

Roman, S.: Advanced Linear Algebra. Springer, New York (1992)

Tian, Y.: Upper and lower bounds for ranks of matrix expressions using generalized inverses. Linear Algebra Appl. 355, 187–214 (2002)

Tian, Y.: The dimension of the intersection of $$k$$ k subspaces. Missouri J. Math. Sci. 14, 1–4 (2002)

Tian, Y.: Equalities and inequalities for ranks of products of generalized inverses of two matrices and their applications. J. Inequal. Appl. 182, 1–51 (2016)

Tian, Y.: How to establish exact formulas for calculating the max-min ranks of products of two matrices and their generalized inverses. Linear Multilinear Algebra 66, 22–73 (2018)

Tian, Y., Jiang, B.: Closed-form formulas for calculating the max–min ranks of a triple matrix product composed by generalized inverses. Comp. Appl. Math. 37, 5876–5919 (2018)

Tian, Y., Styan, G.P.H.: Rank equalities for idempotent and involutory matrices. Linear Algebra Appl. 335, 101–117 (2001)

Von Neumann, J.: Functional Operators. Vol II. The Geometry of Orthogonal Spaces. Princeton University Press, Princeton (1950)

Xiong, Z., Qin, Y.: Invariance properties of an operator product involving generalized inverses. Electron. J. Linear Algebra 22, 694–703 (2011)