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Difference between revisions of "Non-singular matrix"

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''non-degenerate matrix''
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A square [[Matrix|matrix]] with non-zero [[Determinant|determinant]]. For a square matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067300/n0673001.png" /> over a field, non-singularity is equivalent to each of the following conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067300/n0673002.png" /> is invertible; 2) the rows (columns) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067300/n0673003.png" /> are linearly independent; or 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067300/n0673004.png" /> can be brought by elementary row (column) transformations to the identity matrix.
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''non-degenerate matrix''
  
 
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A square [[matrix]] with non-zero [[determinant]]. For a square matrix  $A$
====Comments====
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over a field, non-singularity is equivalent to each of the following conditions: 1)  $A$
 
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is invertible; 2) the rows (columns) of  $A$
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are linearly independent; or 3)  $A$
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can be brought by elementary row (column) transformations to the identity matrix.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Kurosh,  "Matrix theory" , Chelsea, reprint  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.R. McDonald,  "Linear algebra over commutative rings" , M. Dekker  (1984)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.G. Kurosh,  "Matrix theory" , Chelsea, reprint  (1960)  (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  B.R. McDonald,  "Linear algebra over commutative rings" , M. Dekker  (1984)</TD></TR>
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</table>

Latest revision as of 06:01, 23 April 2023


non-degenerate matrix

A square matrix with non-zero determinant. For a square matrix $A$ over a field, non-singularity is equivalent to each of the following conditions: 1) $A$ is invertible; 2) the rows (columns) of $A$ are linearly independent; or 3) $A$ can be brought by elementary row (column) transformations to the identity matrix.

References

[a1] A.G. Kurosh, "Matrix theory" , Chelsea, reprint (1960) (Translated from Russian)
[a2] B.R. McDonald, "Linear algebra over commutative rings" , M. Dekker (1984)
How to Cite This Entry:
Non-singular matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-singular_matrix&oldid=13817
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article