Difference between revisions of "Non-singular matrix"
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| + | ''non-degenerate matrix'' | ||
| + | A square [[Matrix|matrix]] with non-zero [[Determinant|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. | ||
====Comments==== | ====Comments==== | ||
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====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> | <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> | ||
Revision as of 08:03, 6 June 2020
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.
Comments
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
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