<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">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 <italic>m x n</italic> matrix <italic>A</italic> by the condition: <italic>A</italic> has the rank <italic>r</italic> if and only if, there is a <italic>r x r</italic> submatrix of <italic>A</italic> with a non-zero determinant, and for every <italic>k x k</italic> submatrix of <italic>A</italic> with a non-zero determinant we have <italic>k ≤ r.</italic>At the end, I prove that the rank defined by the size of the biggest submatrix with a non-zero determinant of a matrix <italic>A</italic>, is the same as the maximal number of linearly independent rows of <italic>A.</italic></abstract>

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 determinant, and for every k x k submatrix of A with a non-zero determinant we have k ≤ r.At the end, I prove that the rank defined by the size of the biggest submatrix with a non-zero determinant of a matrix A, is the same as the maximal number of linearly independent rows of A.

Basic Properties of the Rank of Matrices over a Field

Institute of Computer Science, University of Białystok, Poland

Formalized Mathematics

{"article-title":"Basic Properties of the Rank of Matrices over a Field"}

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...

Basic Properties of the Rank of Matrices over a Field

Publié en ligne: 09 juin 2008

Pages: 199 - 211

DOI: https://doi.org/10.2478/v10037-007-0024-5

This content is open access.