Idempotency of linear combinations of three idempotent matrices, two of which are disjoint

AbstractGiven nonzero idempotent matrices A1,A2,A3 such that A2 and A3 are disjoint, i.e., A2A3=0=A3A2, the problem of characterizing all situations, in which a linear combination C=c1A1+c2A2+c3A3 is an idempotent matrix, is studied. The results obtained cover those established by J.K. Baksalary, O.M. Baksalary, and G.P.H. Styan (Linear Algebra Appl. 354 (2002) 21) under the additional assumption that c3=−c2, i.e., in the particular case where C=c1A1+c2(A2−A3) is actually a linear combination of an idempotent matrix A1 and a tripotent matrix A2−A3

Some comments on the life and work of Jerzy K. Baksalary (1944-2005)

Following some biographical comments on Jerzy K. Baksalary (1944–2005), this article continues with personal comments by Oskar Maria Baksalary, Tadeusz Cali´nski, R.William Farebrother, Jürgen Groß, Jan Hauke, Erkki Liski, Augustyn Markiewicz, Friedrich Pukelsheim, Tarmo Pukkila, Simo Puntanen, Tomasz Szulc, Yongge Tian, Götz Trenkler, Júlia Volaufová, Haruo Yanai, and Fuzhen Zhang, on the life and work of Jerzy K. Baksalary, and with a detailed list of his publications. Our article ends with a survey by Tadeusz Cali´nski on Jerzy Baksalary’s work in block designs and a set of photographs of Jerzy Baksalary

Further results on generalized and hypergeneralized projectors

AbstractThe notions of generalized and hypergeneralized projectors, introduced by Groß and Trenkler [J. Groß, J. Trenkler, Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474], are revisited. On the one hand, the present paper provides several new characterizations of these sets, and, on the other, the properties of generalized and hypergeneralized projectors related to various matrix partial orderings are considered. Moreover, the paper demonstrates the usefulness, in studying the properties of generalized and hypergeneralized projectors, of the representation of complex matrices given in Corollary 6 by Hartwig and Spindelböck [R.E. Hartwig, K. Spindelböck, Matrices for which A∗ and A† commute, Linear and Multilinear Algebra 14 (1984) 241-256]

Further properties of the star, left-star, right-star, and minus partial orderings

AbstractCertain classes of matrices are indicated for which the star, left-star, right-star, and minus partial orderings, or some of them, are equivalent. Characterizations of the left-star and right-star orderings, similar to those devised by Hartwig and Styan [Linear Algebra Appl. 82 (1986) 145] for the star and minus orderings, are established along with other auxiliary results, which are of independent interest as well. Some inheritance-type properties of matrices are also given. The class of EP matrices appears to be essential in several points of our considerations