Rouché-Frobenius theorem . In linear algebra, this theorem allows us to calculate the type of solutions of a system of linear equations based on the rank of the coefficient matrix and the rank of the extended matrix associated with the system.
Summary
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- 1 Story
- 2 Definition
- 3 Sources
- 4 External links
History
It is named after the French mathematician Eugèn Rouché who enunciated it in 1875 and the German mathematician Ferdinand Georg Frobenius who was one of the many mathematicians who proved it. The mathematician Georges Fontené claimed authorship of the proof of the theorem, and later, in 1905 , Frobenius credited the authorship to both Rouché and Fontené.
The theorem is known in Russia as the Kronecker-Capelli Theorem ; in Italy , as the Rocuhé-Frobenius Theorem ; and, in France , as Rouché-Fontené’s theorem .
Definition
The theorem establishes that for a system of linear equations to be compatible it is a necessary and sufficient condition that the matrix formed by coefficients and the expanded one have the same range. In addition, the system will be determined if its rank coincides with the number of unknowns or it will be indeterminate if it has a value less than that number.
