Thomas Ernst (2015)
Special Matrices
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In this second article on q-Pascal matrices, we show how the previous factorizations by the summation matrices and the so-called q-unit matrices extend in a natural way to produce q-analogues of Pascal matrices of two variables by Z. Zhang and M. Liu as follows [...] We also find two different matrix products for [...]
Jiří Rohn (1990)
Aplikace matematiky
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New proofs of two previously published theorems relating nonsingularity of interval matrices to -matrices are given.
Waldemar Hołubowski (2002)
Discussiones Mathematicae - General Algebra and Applications
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In this note we explain why the group of n×n upper triangular matrices is defined usually over commutative ring while the full general linear group is defined over any associative ring.
Nobuyuki Tamura, Yatsuka Nakamura (2007)
Formalized Mathematics
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In this paper the classic theory of matrices of real elements (see e.g. [12], [13]) is developed. We prove selected equations that have been proved previously for matrices of field elements. Similarly, we introduce in this special context the determinant of a matrix, the identity and zero matrices, and the inverse matrix. The new concept discussed in the case of matrices of real numbers is the property of matrices as operators acting on finite sequences of real numbers from both sides....
Yatsuka Nakamura, Nobuyuki Tamura, Wenpai Chang (2006)
Formalized Mathematics
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Here, the concept of matrix of real elements is introduced. This is defined as a special case of the general concept of matrix of a field. For such a real matrix, the notions of addition, subtraction, scalar product are defined. For any real finite sequences, two transformations to matrices are introduced. One of the matrices is of width 1, and the other is of length 1. By such transformations, two products of a matrix and a finite sequence are defined. Also the linearity of such product...
Miroslav Fiedler, Vlastimil Pták (1962)
Czechoslovak Mathematical Journal
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M. Rajesh Kannan, K.C. Sivakumar (2014)
Discussiones Mathematicae - General Algebra and Applications
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Let A and B be M-matrices satisfying A ≤ B and J = [A,B] be the set of all matrices C such that A ≤ C ≤ B, where the order is component wise. It is rather well known that if A is an M-matrix and B is an invertible M-matrix and A ≤ B, then aA + bB is an invertible M-matrix for all a,b > 0. In this article, we present an elementary proof of a stronger version of this result and study corresponding results for certain other classes as well.