# zbMATH — the first resource for mathematics

Nonnegative and positive definiteness of matrices modified by two matrices of rank one. (English) Zbl 0728.15011
From the authors’ summary: Necessary and sufficient conditions are given for the nonnegative and positive definiteness of matrices of the form $$A- a_ 1a^*_ 1-a_ 2a^*_ 2$$ and $$A+a_ 1a^*_ 1-a_ 2a^*_ 2$$, where A is a Hermitian matrix and $$a_ 1,a_ 2$$ are complex vectors.

##### MSC:
 15B57 Hermitian, skew-Hermitian, and related matrices 15B48 Positive matrices and their generalizations; cones of matrices
Full Text:
##### References:
 [1] Albert, A., Conditions for positive and nonnegative definiteness in terms of pseudo-inverses, SIAM J. appl. math., 17, 434-440, (1969) · Zbl 0265.15002 [2] Baksalary, J.K; Kala, R., Partial orderings between matrices one of which is of rank one, Bull. Polish acad. sci. math., 31, 5-7, (1983) · Zbl 0535.15006 [3] Baksalary, J.K.; Liski, E.P.; Trenkler, G., Mean square error matrix improvements and admissibility of linear estimators, J. statist. plann. inference, 23, 313-325, (1989) · Zbl 0685.62052 [4] Baksalary, J.K.; Pukelsheim, F., A note on the matrix ordering of special C-matrices, Linear algebra appl., 70, 263-267, (1985) · Zbl 0603.62071 [5] Baksalary, J.K.; Puri, P.D., Criteria for the validity of Fisher’s condition for balanced block designs, J. statist. plann. inference, 18, 119-123, (1988) · Zbl 0663.62079 [6] Bekker, P.A., The positive semidefiniteness of partitioned matrices, Linear algebra appl., 111, 261-278, (1988) · Zbl 0658.15021 [7] Campbell, S.L.; Meyer, C.D., Generalized inverses of linear transformations, (1979), Pitman London · Zbl 0417.15002 [8] Caron, R.J.; Gould, N.I.M., Finding a positive semidefinite interval for a parametric matrix, Linear algebra appl., 76, 19-29, (1986) · Zbl 0593.15015 [9] Christof, K.; Pukelsheim, F., Approximate design theory for a simple block design with random block effects, (), 20-28 · Zbl 0593.62073 [10] Farebrother, R.W., Further results on the Mean square error of ridge regression, J. roy. statist. soc. ser. B, 38, 248-250, (1976) · Zbl 0344.62056 [11] Farebrother, R.W., Three theorems with applications to Euclidean distance matrices, Linear algebra appl., 95, 11-16, (1987) · Zbl 0627.15008 [12] Meyer, C.D., Generalized inversion of modified matrices, SIAM J. appl. math., 24, 315-323, (1973) · Zbl 0253.15001 [13] Rao, C.R.; Mitra, S.K., Generalized inverse of matrices and its applications, (1971), Wiley New York [14] Trenkler, G., Mean square error matrix comparisons of estimators in linear regression, Comm. statist. theory methods, 14, 2495-2509, (1985) · Zbl 0594.62075 [15] Trenkler, G., Mean square error matrix comparisons among restricted least squares estimators, Sankhyā ser. A, 49, 96-104, (1987) · Zbl 0639.62060
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.