Evaluating 



AT,S(2)
A_{T,S}^{\left( 2 \right)}


 Matrix Inverses from full-rank Singular Value Decomposition

Stanimirović, Ivan

doi:10.2478/auom-2025-0023

Abstract

Based on the compact SVD decomposition of the full rank of a predetermined matrix M, a method for calculating the inverses $A_{T, S}^{(2)}$ A_{T,S}^{\left( 2 \right)} of matrix A is derived. As a consequence, the advantages of compact single value decompositions were developed with this new method. The method is then extended to a set of polynomial matrices. As a result, an algorithm is proposed for effective symbolic computation of $A_{T, S}^{(2)}$ A_{T,S}^{\left( 2 \right)} inverses of a polynomial matrix. Some implementation details and comparative computation times compared to other similar methods are shown to illustrate the efficiency of the algorithm.

References

A. G. Akritas, G. I. Malaschonok, Computations in Modules over Commutative Domains, Lecture Notes in Computer Science, 4770, (2007), 11–23.
Search in Google Scholar Back to article
A. Ben-Israel, T.N.E. Greville, Generalized inverses: Theory and Applications, Second Ed., Springer, 2003.
Search in Google Scholar Back to article
Y. Chen, The generalized Bott–Duffin inverse and its application, Linear Algebra Appl. 134 (1990), 71–91.
Search in Google Scholar Back to article
P. Courrieu, Fast solving of weighted pairing least-squares systems, Journal of Computational and Applied Mathematics 231 (2009), 39–48.
Search in Google Scholar Back to article
P. Courrieu, Fast Computation of Moore-Penrose Inverse Matrices, Neural Information Processing - Letters and Reviews 8 (2005), 25–29.
Search in Google Scholar Back to article
J. Foster, J. McWhirter, M. Davies, J. Chambers, An algorithm for calculating the QR and singular value decompositions of polynomial matrices. IEEE Transactions on Signal Processing, 58(3), (2009) 1263–1274.
Search in Google Scholar Back to article
J. Foster, Algorithms and techniques for polynomial matrix decompositions, Ph.D. dissertation, School Eng., Cardiff Univ., Cardiff, U.K., 2008.
Search in Google Scholar Back to article
J. Jones, N.P. Karampetakis and A.C. Pugh, The computation and application of the generalized inverse via Maple, J. Symbolic Comput. 25 (1998), 99–124.
Search in Google Scholar Back to article
A.N. Malyshev, ”Matrix equations: Factorization of matrix polynomials” M. Hazewinkel (ed.), Handbook of Algebra I, Elsevier (1995), 79–116.
Search in Google Scholar Back to article
[10] X. Sheng, G. Chen, Full-rank representation of generalized inverse AT,S(2) A_{T,S}^{\left( 2 \right)} and its applications, Comput. Math. Appl. 54 (2007), 1422–1430.
Back to article
I.P. Stanimirović, M.B. Tasić, Computation of generalized inverses by using the LDL* decomposition, Appl. Math. Lett., 25 (2012), 526–531.
Search in Google Scholar Back to article
P.S. Stanimirović and M.B. Tasić, Partitioning method for rational and polynomial matrices, Appl. Math. Comput. 155 (2004), 137–163.
Search in Google Scholar Back to article
M. Tasić, I. Stanimirović, Symbolic computation of Moore-Penrose inverse using the LDL* decomposition of the polynomial matrix, Filomat 27:8 (2013), 1393–1403.
Search in Google Scholar Back to article
M.B. Tasić, P.S. Stanimirović, M.D. Petković, Symbolic computation of weighted Moore-Penrose inverse using partitioning method, Appl. Math. Comput. 189 (2007), 615–640.
Search in Google Scholar Back to article
M.B. Tasić, P.S. Stanimirović, Symbolic and recursive computation of different types of generalized inverses, Appl. Math. Comput. 199 (2008), 349–367.
Search in Google Scholar Back to article
M.B. Tasić, P.S. Stanimirović, Differentiation of generalized inverses for rational and polynomial matrices, Appl. Math. Comput. 216 (2010), 2092–2106.
Search in Google Scholar Back to article
G. Wang, Y. Wei, S. Qiao, Generalized Inverses: Theory and Computations, Science Press, Beijing/New York, 2004.
Search in Google Scholar Back to article
Y. Wei, C. Deng, A note on additive results for the Drazin inverse, Linear and Multilinear Algebra 59:12 (2011), 1319–1329.
Search in Google Scholar Back to article
S. Wolfram, The mathematica Book, 5th ed., Wolfram Media/Cambridge University Press, Champaign, IL 61820, USA, 2003.
Search in Google Scholar Back to article
G. Zielke, Report on test matrices for generalized inverses, Computing 36 (1986), 105–162.
Search in Google Scholar Back to article

Evaluating AT,S(2) A_{T,S}^{\left( 2 \right)} Matrix Inverses from full-rank Singular Value Decomposition

Abstract

Paradigm

My account