Displaying similar documents to “Determinant of Some Matrices of Field Elements”

Laplace Expansion

Karol Pak, Andrzej Trybulec (2007)

Formalized Mathematics

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In the article the formula for Laplace expansion is proved.

A Theory of Matrices of Real Elements

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...

Determinant and Inverse of Matrices of Real Elements

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....

Basic Properties of Determinants of Square Matrices over a Field 1

Karol Pąk (2007)

Formalized Mathematics

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In this paper I present basic properties of the determinant of square matrices over a field and selected properties of the sign of a permutation. First, I define the sign of a permutation by the requirement [...] where p is any fixed permutation of a set with n elements. I prove that the sign of a product of two permutations is the same as the product of their signs and show the relation between signs and parity of permutations. Then I consider the determinant of a linear combination...

Basic Properties of the Rank of Matrices over a Field

Karol Pąk (2007)

Formalized Mathematics

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In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field.I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the concept of the rank of a m x n matrix A by the condition: A has the rank r if and only if, there is a r x r submatrix of A with a non-zero...

Generalizations of Nekrasov matrices and applications

Ljiljana Cvetković, Vladimir Kostić, Maja Nedović (2015)

Open Mathematics

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In this paper we present a nonsingularity result which is a generalization of Nekrasov property by using two different permutations of the index set. The main motivation comes from the following observation: matrices that are Nekrasov matrices up to the same permutations of rows and columns, are nonsingular. But, testing all the permutations of the index set for the given matrix is too expensive. So, in some cases, our new nonsingularity criterion allows us to use the results already...

Explicit formulas for the constituent matrices. Application to the matrix functions

R. Ben Taher, M. Rachidi (2015)

Special Matrices

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We present a constructive procedure for establishing explicit formulas of the constituents matrices. Our approach is based on the tools and techniques from the theory of generalized Fibonacci sequences. Some connections with other results are supplied. Furthermore,we manage to provide tractable expressions for the matrix functions, and for illustration purposes we establish compact formulas for both the matrix logarithm and the matrix pth root. Some examples are also provided. ...