×

zbMATH — the first resource for mathematics

Three results in connection with inverse matrices. (English) Zbl 0562.15003
i. The Moore-Penrose solution of an arbitrary system of linear equations is a convex combination of the solutions of all uniquely solvable partial systems. ii. For a determinant appearing in an explicit formula of D. S. Meek [ibid. 49, 117-129 (1983; Zbl 0505.15005)] concerning the elements of the inverse of a Toeplitz band matrix the asymptotic representation is established under general conditions. iii. For the limit case of an explicit formula of W. D. Hoskins and P. J. Ponzo [Math. Comput. 26, 393-400 (1972; Zbl 0248.15008)] concerning the elements of the inverse of a Toeplitz band matrix with special binomial coefficients there is given a modification and generalization. In the last two cases there are used methods from the discrete operational calculus.

MSC:
15A09 Theory of matrix inversion and generalized inverses
44A55 Discrete operational calculus
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berg, L., Asymptotische abschätzung inverser matrizen mit einer anwendung auf partielle differentialgleichungen, Z. angew. math. mech., 60, 453-458, (1980) · Zbl 0424.35002
[2] Berg, L., Die invertierung von matrizen aus binomialkoeffizienten, Z. angew. math. mech., 63, 639-642, (1983) · Zbl 0497.15009
[3] Berg, L., Asymptotische abschätzung der inversen toeplitzcher bandmatrizen im grenzfall, Z. anal. anwendungen, 3, 179-191, (1984) · Zbl 0512.41024
[4] Berg, L., Über die greensche funktion und die reduzierte wronskische determinante, Rostock. math. kolloq., 26, 45-50, (1984) · Zbl 0551.39003
[5] L. Berg, Lineare Gleichungssysteme mit Bandstruktur und ihr asymptotisches Verhalten, VEB DVW, Berlin, to appear · Zbl 0613.15004
[6] Berg, L., Über eine identität von W.F. trench zwischen der toeplitzschen und einer verallgemeinerten vandermondeschen determinante, Z. angew. math. mech., 66, 314-315, (1986) · Zbl 0557.15005
[7] Berg, L., On the majorization method for holomorphic solutions of linear partial differential equations, Z. anal. anwendungen, 5, 111-117, (1986) · Zbl 0556.35019
[8] Böttcher, A.; Silbermann, B., Invertibility and asymptotics of Toeplitz matrices, () · Zbl 0578.47015
[9] Brezinski, C., Some determinantal identities in a vector space with applications, (), 1-11
[10] Gabriel, R., Pseudoinversen mit schlüssel und ein system der algebraischen kryptographie, Rev. roumaine math. pures appl., 22, 1077-1099, (1977) · Zbl 0398.15006
[11] Heinig, G.; Rost, K., Algebraic methods for Toeplitz-like matrices and operators, () · Zbl 1084.65030
[12] Hoskins, W.D.; Ponzo, P.J., Some properties of a class of band matrices, Math. comp., 26, 393-400, (1972) · Zbl 0248.15008
[13] Meek, D.S., The inverses of Toeplitz band matrices, Linear algebra appl., 49, 117-129, (1983) · Zbl 0505.15005
[14] Moore, E.H., On the reciprocal of the general algebraic matrix, abstract, Bull. amer. math. soc., 26, 394-395, (1920)
[15] Smirnow, W.I., (), (transl. from Russian) · Zbl 0997.26502
[16] Springer, J., Exakte rechnung durch residuenarithmetik und einige Möglichkeiten ihrer anwendung, ()
[17] Springer, J., Die exakte berechnung der Moore-Penrose-inversen einer matrix durch residuenarithmetik, Z. angew. math. mech., 63, 203-210, (1983) · Zbl 0532.65024
[18] Stojaković, M., Generalized inverse matrices, Mathematical structures—computational mathematics—mathematical modelling, 461-470, (1975), Sofia
[19] Trench, W.F., On the eigenvalue problem for Toeplitz band matrices, Linear algebra appl., 64, 199-214, (1985) · Zbl 0568.15005
[20] Verghese, G.C., A “cramer rule” for the least-norm, least-squared-error solution of inconsistent linear equations, Linear algebra appl., 48, 315-316, (1982) · Zbl 0501.15004
[21] Werner, H.J., On extensions of Cramer’s rule for solutions of restricted linear systems, Linear and multilinear algebra, 15, 319-330, (1984) · Zbl 0544.15002
[22] Zielke, G., Verallgemeinerte inverse matrizen, (), 95-116 · Zbl 0531.15002
[23] G. Zielke, Private communication, 3 Mar. 1985.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.