Basic Properties of the Rank of Matrices over a Field

Karol Pąk

Formalized Mathematics (2007)

  • Volume: 15, Issue: 4, page 199-211
  • ISSN: 1426-2630

Abstract

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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 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.

How to cite

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Karol Pąk. "Basic Properties of the Rank of Matrices over a Field." Formalized Mathematics 15.4 (2007): 199-211. <http://eudml.org/doc/267151>.

@article{KarolPąk2007,
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.},
author = {Karol Pąk},
journal = {Formalized Mathematics},
language = {eng},
number = {4},
pages = {199-211},
title = {Basic Properties of the Rank of Matrices over a Field},
url = {http://eudml.org/doc/267151},
volume = {15},
year = {2007},
}

TY - JOUR
AU - Karol Pąk
TI - Basic Properties of the Rank of Matrices over a Field
JO - Formalized Mathematics
PY - 2007
VL - 15
IS - 4
SP - 199
EP - 211
AB - 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.
LA - eng
UR - http://eudml.org/doc/267151
ER -

References

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Citations in EuDML Documents

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  1. Xiaopeng Yue, Xiquan Liang, Basic Properties of Circulant Matrices and Anti-Circular Matrices
  2. Xiquan Liang, Tao Wang, Some Basic Properties of Some Special Matrices. Part III
  3. Karol Pąk, Block Diagonal Matrices
  4. Karol Pąk, Solutions of Linear Equations
  5. Karol Pąk, Eigenvalues of a Linear Transformation
  6. Karol Pąk, Linear Transformations of Euclidean Topological Spaces. Part II
  7. Karol Pąk, Linear Map of Matrices
  8. Karol Pąk, Linear Transformations of Euclidean Topological Spaces
  9. Karol Pąk, The Rotation Group

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