Kronecker-Capelli's theorem

The set of Kronecker Capelli is releasably criterion for systems of linear equations . It is named after the mathematicians Leopold Kronecker (1823-1891) and Alfredo Capelli (1855-1910), but was previously used in various formulations by other mathematicians, including Fontené , Rouché and Frobenius . Accordingly, the sentence often has different names in (international) literature, is simply referred to as the solvability criterion or is used without a name.

statement

To a system of linear equations

${\displaystyle {\begin{matrix}a_{11}x_{1}+a_{12}x_{2}\,+&\cdots &+\,a_{1n}x_{n}&=&b_{1}\\a_{21}x_{1}+a_{22}x_{2}\,+&\cdots &+\,a_{2n}x_{n}&=&b_{2}\\&&&\vdots &\\a_{m1}x_{1}+a_{m2}x_{2}\,+&\cdots &+\,a_{mn}x_{n}&=&b_{m}\\\end{matrix}}}$

denote its coefficient matrix ${\displaystyle A}$

${\displaystyle {\begin{pmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\end{pmatrix}}}$

and its extended coefficient matrix ${\displaystyle (A|b)}$

${\displaystyle \left({\begin{array}{cccc|c}a_{11}&a_{12}&\cdots &a_{1n}&b_{1}\\a_{21}&a_{22}&\cdots &a_{2n}&b_{2}\\\vdots &\vdots &\ddots &\vdots &\\a_{m1}&a_{m2}&\cdots &a_{mn}&b_{m}\end{array}}\right)}$

Kronecker-Capelli's theorem now states that this system of equations has (at least) one solution if and only if the rank of the coefficient matrix corresponds to the rank of the extended coefficient matrix, i.e. ${\displaystyle A}$ ${\displaystyle (A|b)}$

${\displaystyle {\text{rang}}(A)={\text{rang}}(A|b)}$

applies.

literature

• Kronecker-Capelli theorem in the Encyclopaedia of Mathematics
• Andreas Filler: Elementary Linear Algebra: Linearizing and Coordinating . Springer, 2011, ISBN 9783827424136 , pp. 34-40
• Georgi E. Shilov, Richard A. Silverman: An Introduction to the Theory of Linear Spaces . Courier (Dover), 2012, ISBN 9780486139432 , pp. 54-55

Web links

• Kronecker-Capelli theorem on Wikibooks
• Michael Drmota: Linear Algebra I . Script, TU Wien, 2005, p. 70, sentence 4.69

Individual evidence

1. ↑ ^{a b} Andreas Filler: Elementary Linear Algebra: Linearizing and Coordinating . Springer, 2011, ISBN 9783827424136 , pp. 34-40
2. ^ Kronecker-Capelli in the Encyclopaedia of Mathematics
3. ↑ Gerd Fischer: Linear Algebra . Vieweg, 9th edition, 1989, p. 125