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Cayley-Hamilton Theorem

 
 

Consider the ~n # ~n matrix A over an arbitrary field F. _ @M_{~n}(F), the set of all ~n # ~n matrices over F, is a vector space of dimension ~n. Therefore the ~n^2+1 matrices I, A, A^2, ... A^{~n # ~n} must be linearly dependent, so &exist. &alpha._2, ... &alpha._{~n # ~n} such that:

sum{ &alpha._{~i} A^{~i},~i = 0,~n^2} _ _ = _ _ 0

Thus every square matrix satisfies a polynomial of degree ~k , ~k &le. ~n^2. _ However an even more remarkable result holds:

Cayley-Hamilton Theorem

Let

f ( λ ) _ = _ det{A &minus. λI } _ = _ sum{p_{~i} λ^{~n&minus.~i},~i = 0,~n}

be the characteristic polynomial of A. _ Then _ f ( A ) = 0, _ i.e. every matrix satisfies its own characteristic equation.