Difference between revisions of "Non-singular matrix"
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''non-degenerate matrix'' | ''non-degenerate matrix'' | ||
| − | A square [[ | + | 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) $ | + | over a field, non-singularity is equivalent to each of the following conditions: 1) $A$ |
| − | is invertible; 2) the rows (columns) of $ | + | is invertible; 2) the rows (columns) of $A$ |
| − | are linearly independent; or 3) $ | + | are linearly independent; or 3) $A$ |
can be brought by elementary row (column) transformations to the identity matrix. | 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> | + | <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> | ||
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=53854
Non-singular matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-singular_matrix&oldid=53854
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article