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'''[[Eugène Rouché|Rouché]]–[[Alfredo Capelli|Capelli]] theorem''' is the theorem in [[linear algebra]] that allows computing the number of solutions in a [[system of linear equations]] given the ranks of its [[augmented matrix]] and [[coefficient matrix]]. The theorem is known as '''[[Leopold Kronecker|Kronecker]]–Capelli theorem''' in Russia, '''Rouché–Capelli theorem''' in Italy, '''Rouché–Fontené theorem''' in France and '''Rouché–[[Ferdinand Georg Frobenius|Frobenius]]''' theorem in Spain and many countries in Latin America.
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== Formal statement ==
A system of linear equations with n variables has a solution [[if and only if]] the [[Rank (linear algebra)|rank]] of its [[coefficient matrix]] ''A'' is equal to the rank of its [[augmented matrix]]&nbsp;[''A''|''b'']. If there are solutions, they form an [[affine subspace]] of <math>\mathbb{R}^n</math> of dimension ''n''&nbsp;&minus;&nbsp;rank(''A''). In particular:
* if ''n''&nbsp;=&nbsp;rank(''A''), the solution is unique,
* otherwise there are infinite number of solutions.
 
==References==
* {{cite book | author=A. Carpinteri | title=Structural mechanics | page=74 | publisher=Taylor and Francis | isbn=0-419-19160-7 | year=1997 }}
 
{{DEFAULTSORT:Rouche-Capelli theorem}}
[[Category:Theorems in linear algebra]]
[[Category:Matrix theory]]
 
 
{{Linear-algebra-stub}}
 
[[cs:Soustava lineárních rovnic#Frobeniova věta]]

Latest revision as of 08:12, 6 May 2014

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