×

zbMATH — the first resource for mathematics

On the Wedderburn-Guttman theorem. (English) Zbl 1111.15001
The paper deals with some extension on the Wedderburn-Guttman theorem [cf. J. H. M. Wedderburn, Lectures on matrices (1934; republ. 1964; Zbl 0121.26101); L. Guttman, Psychometrika 9, 1-16 (1944; Zbl 0060.31212)]. This theorem says that if \(A\) is an \(m \times n\) matrix of rank \(a\), \(M\) and \(N\) matrices of sizes \(m \times p\) and \(n \times p\), respectively, such that \(M^TAN\) is nonsingular, then \[ \text{rank}(A-AN(M^TAN)^{-1}M^TA)=a-p, \] where \(p=\text{rank}(AN(M^TAN)^{-1}M^TA)= \text{rank}(M^TAN)\).
In this paper the authors focus on the case in which \(M^TAN\) is rectangular or singular. Let \(M\) and \(N\) be \(m \times r\) and \(n \times s\) matrices, respectively, where \(r\) is not necessarily equal to \(s\), or \(\text{rank}(M^TAN)< min \{r,s\}\). The authors investigate conditions under which the regular inverse \((M^TAN)^{-1}\) can be replaced by a \(g\)-inverse \((M^TAN)^{-}\) of some kind, thereby extending the Wedderburn-Guttman theorem. Of course, in this case \(\text{rank}(AN(M^TAN)^{-}M^TA)=g\) may not be equal to \(\text{rank}(M^TAN)=h\). Here, the authors study necessary and sufficient condicitons in order to \(\text{rank}(A-AN(M^TAN)^{-}M^TA)=a-g\). The resultant conditions look similar to those arising in seemingly unrelated contexts, namely Cochran’s and related theorems [cf. P. Šemrl, Linear Algebra Appl. 237–238, 477–487 (1996; Zbl 0847.62043), Section IV] on distributions of quadratic forms involving a normal random vector.

MSC:
15A03 Vector spaces, linear dependence, rank, lineability
15A09 Theory of matrix inversion and generalized inverses
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, T.W.; Styan, G.P.H., Cochran’s theorem, rank additivity and tripotent matrices, (), 1-23 · Zbl 0505.62030
[2] Baksalary, J.K.; Hauke, J., A further algebraic version of cochran’s theorem and matrix partial orderings, Linear algebra appl., 127, 157-169, (1990) · Zbl 0696.15005
[3] Baksalary, J.K.; Kala, R., Range invariance of certain matrix products, Linear multilinear algebra, 14, 89-96, (1983) · Zbl 0523.15006
[4] Baksalary, J.K.; Mathew, T., Rank invariance criterion and its application to the unified theory of least squares, Linear algebra appl., 127, 393-401, (1990) · Zbl 0694.15003
[5] Ben-Israel, A.; Greville, T.N.E., Generalized inverse, (1974), Wiley New York
[6] Chu, M.T.; Funderlic, R.E.; Golub, G.H., A rank-one reduction formula and its applications to matrix factorizations, SIAM rev., 37, 512-530, (1995) · Zbl 0844.65033
[7] Cline, R.E.; Funderlic, R.E., The rank of a difference of matrices and associated generalized inverses, Linear algebra appl., 24, 185-215, (1979) · Zbl 0393.15005
[8] Groß, J., Comment on range invariance of matrix products, Linear multilinear algebra, 41, 157-160, (1996) · Zbl 0871.15003
[9] Guttman, L., General theory and methods of matric factoring, Psychometrika, 9, 1-16, (1944) · Zbl 0060.31212
[10] Guttman, L., Multiple group methods for common-factor analysis: their basis, computation and interpretation, Psychometrika, 17, 209-222, (1952) · Zbl 0049.37503
[11] Guttman, L., A necessary and sufficient formula for matric factoring, Psychometrika, 22, 79-81, (1957) · Zbl 0080.13202
[12] Hartwig, R.E., How to partially order regular elements, Math. japon., 25, 1-13, (1980) · Zbl 0442.06006
[13] Hartwig, R.E., A note on rank-additivity, Linear multilinear algebra, 10, 59-61, (1981) · Zbl 0456.15002
[14] Hartwig, R.E.; Styan, G.P.H., On some characterizations of “star” partial ordering for matrices and rank subtractivity, Linear algebra appl., 82, 145-161, (1986) · Zbl 0603.15001
[15] Horst, P., Matrix algebra for social scientists, (1965), Holt, Rinehart and Winston New York · Zbl 0121.26103
[16] Householder, A.S., The theory of matrices in numerical analysis, (1964), Blaisdell New York · Zbl 0161.12101
[17] Hubert, L.; Meulman, J.; Heiser, W., Two purposes for matrix factorization: a historical appraisal, SIAM rev., 42, 68-82, (2000) · Zbl 0999.65014
[18] Jain, S.K.; Mitra, S.K.; Werner, H.J., Extensions of \(\mathcal{G}\)-based matrix partial order, SIAM J. matrix anal. appl., 17, 834-850, (1996) · Zbl 0860.15007
[19] Khatri, C.G., A note on a MANOVA model applied to problems in growth curves, Ann. inst. statist. math., 18, 75-86, (1966) · Zbl 0136.40704
[20] Marsaglia, G.; Styan, G.P.H., When does rank(A+B)=rank(A)+rank(B)?, Can. math. bull., 15, 451-452, (1972) · Zbl 0252.15002
[21] Marsaglia, G.; Styan, G.P.H., Equalities and inequalities for ranks of matrices, Linear multilinear algebra, 2, 269-292, (1974)
[22] Mitra, S.K., A new class of g-inverse of square matrices, Sankhyā ser. A, 30, 322-330, (1968) · Zbl 0198.35104
[23] Mitra, S.K., Fixed rank solutions of linear matrix equations, Sankhyā ser. A, 30, 387-392, (1972) · Zbl 0261.15008
[24] Mitra, S.K., The minus partial order and the shorted matrix, Linear algebra appl., 83, 1-27, (1986) · Zbl 0605.15004
[25] Ogasawara, T.; Takahashi, M., Independence of quadratic forms of a random sample from a normal population, Sci. bull. Hiroshima univ., 15, 1-9, (1951)
[26] Rao, C.R., Linear statistical inference and its applications, (1973), Wiley New York · Zbl 0169.21302
[27] Rao, C.R.; Mitra, S.K., Generalized inverse of matrices and its applications, (1971), Wiley New York
[28] Schönemann, P.H.; Steiger, J.H., Regression component analysis, British J. math. statist. psych., 29, 175-189, (1976) · Zbl 0356.92026
[29] Šemrl, P., On a matrix version of cochran’s statistical theorem, Linear algebra appl., 237-238, 477-487, (1996) · Zbl 0847.62043
[30] Shanbag, D.N., Some remarks on khatri’s result in quadratic forms, Biometrika, 55, 593-595, (1968) · Zbl 0177.47403
[31] Shanbag, D.N., On the distribution of a quadratic form, Biometrika, 57, 222-223, (1970) · Zbl 0193.18101
[32] Styan, G.P.H., Notes on the distribution of quadratic forms in singular normal variables, Biometrika, 57, 567-572, (1970) · Zbl 0264.62006
[33] Takane, Y.; Hunter, M.A., Constrained principal component analysis: a comprehensive theory, Appl. algebra engrg. comm. comput., 12, 391-419, (2001) · Zbl 1040.62050
[34] Y. Tian, G.P.H. Styan, On the relationship between two matrix sets A− and PN−Q of generalized inverses, Unpublished manuscript, Queens University, 2004.
[35] Wedderburn, J.H.M., Lectures on matrices, Colloquium publications, vol. 17, (1934), American Mathematical Society Providence, and Dover, New York, 1964 · Zbl 0010.09904
[36] Werner, H.J., Generalized inversion and weak bi-complementarity, Linear multilinear algebra, 19, 357-372, (1986) · Zbl 0598.15005
[37] Yanai, H., Some generalized forms of least squares g-inverse, minimum norm g-inverse and Moore-Penrose g-inverse matrices, Comput. statist. data anal., 10, 251-260, (1990) · Zbl 0825.62550
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.