Difference between revisions of "Kronecker-Capelli theorem"
From Encyclopedia of Mathematics
Ulf Rehmann (talk | contribs) m (moved Kronecker–Capelli theorem to Kronecker-Capelli theorem: ascii title) |
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A system of equations | A system of equations | ||
| + | $$ | ||
| + | \begin{array}{ccc} | ||
| + | a_{11} x_1 + \cdots + a_{1n}x_n &=& b_1 \\ | ||
| + | \vdots & \vdots & \vdots \\ | ||
| + | a_{n1} x_1 + \cdots + a_{nn}x_n &=& b_n | ||
| + | \end{array} | ||
| + | $$ | ||
| − | + | is compatible if and only if the [[Rank|rank]] of the coefficient matrix $A = (a_{ij})$ is equal to that of the augmented matrix $\bar A$ obtained from $A$ by adding the column of free terms $b_i$. | |
| − | |||
| − | is compatible if and only if the [[Rank|rank]] of the coefficient matrix | ||
Kronecker's version of this theorem is contained in his lectures read at the University of Berlin in 1883–1891 (see [[#References|[1]]]). A. Capelli was apparently the first to state the theorem in the above form, using the term "rank of a matrix" (see [[#References|[2]]]). | Kronecker's version of this theorem is contained in his lectures read at the University of Berlin in 1883–1891 (see [[#References|[1]]]). A. Capelli was apparently the first to state the theorem in the above form, using the term "rank of a matrix" (see [[#References|[2]]]). | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> L. Kronecker, "Vorlesungen über die Theorie der Determinanten" , Leipzig (1903)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Capelli, "Sopra la compatibilitá o incompatibilitá di più equazioni di primo grado fra picì incognite" ''Revista di Matematica'' , '''2''' (1892) pp. 54–58</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> L. Kronecker, "Vorlesungen über die Theorie der Determinanten" , Leipzig (1903)</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> A. Capelli, "Sopra la compatibilitá o incompatibilitá di più equazioni di primo grado fra picì incognite" ''Revista di Matematica'' , '''2''' (1892) pp. 54–58</TD></TR> | ||
| + | <TR><TD valign="top">[3]</TD> <TD valign="top"> A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian)</TD></TR> | ||
| + | </table> | ||
| + | |||
| + | {{TEX|done}} | ||
Revision as of 18:12, 26 October 2014
compatibility criterion for a system of linear equations
A system of equations $$ \begin{array}{ccc} a_{11} x_1 + \cdots + a_{1n}x_n &=& b_1 \\ \vdots & \vdots & \vdots \\ a_{n1} x_1 + \cdots + a_{nn}x_n &=& b_n \end{array} $$
is compatible if and only if the rank of the coefficient matrix $A = (a_{ij})$ is equal to that of the augmented matrix $\bar A$ obtained from $A$ by adding the column of free terms $b_i$.
Kronecker's version of this theorem is contained in his lectures read at the University of Berlin in 1883–1891 (see [1]). A. Capelli was apparently the first to state the theorem in the above form, using the term "rank of a matrix" (see [2]).
References
| [1] | L. Kronecker, "Vorlesungen über die Theorie der Determinanten" , Leipzig (1903) |
| [2] | A. Capelli, "Sopra la compatibilitá o incompatibilitá di più equazioni di primo grado fra picì incognite" Revista di Matematica , 2 (1892) pp. 54–58 |
| [3] | A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) |
How to Cite This Entry:
Kronecker-Capelli theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kronecker-Capelli_theorem&oldid=22669
Kronecker-Capelli theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kronecker-Capelli_theorem&oldid=22669
This article was adapted from an original article by I.V. Proskuryakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article