Displaying similar documents to “Explicit formulas for the constituent matrices. Application to the matrix functions”

A Theory of Matrices of Real Elements

Yatsuka Nakamura, Nobuyuki Tamura, Wenpai Chang (2006)

Formalized Mathematics

Similarity:

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...

Factorizations for q-Pascal matrices of two variables

Thomas Ernst (2015)

Special Matrices

Similarity:

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 [...]

Determinant and Inverse of Matrices of Real Elements

Nobuyuki Tamura, Yatsuka Nakamura (2007)

Formalized Mathematics

Similarity:

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....

Intervals of certain classes of Z-matrices

M. Rajesh Kannan, K.C. Sivakumar (2014)

Discussiones Mathematicae - General Algebra and Applications

Similarity:

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.

On the matrix form of Kronecker lemma

João Lita da Silva, António Manuel Oliveira (2009)

Discussiones Mathematicae Probability and Statistics

Similarity:

A matrix generalization of Kronecker's lemma is presented with assumptions that make it possible not only the unboundedness of the condition number considered by Anderson and Moore (1976) but also other sequences of real matrices, not necessarily monotone increasing, symmetric and nonnegative definite. A useful matrix decomposition and a well-known equivalent result about convergent series are used in this generalization.

Nonsingularity and P -matrices.

Jiří Rohn (1990)

Aplikace matematiky

Similarity:

New proofs of two previously published theorems relating nonsingularity of interval matrices to P -matrices are given.

Some Special Matrices of Real Elements and Their Properties

Xiquan Liang, Fuguo Ge, Xiaopeng Yue (2006)

Formalized Mathematics

Similarity:

This article describes definitions of positive matrix, negative matrix, nonpositive matrix, nonnegative matrix, nonzero matrix, module matrix of real elements and their main properties, and we also give the basic inequalities in matrices of real elements.