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Mathematics
Linear and Abstract Algebra
Prove that the limit of this matrix expression is 0
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[QUOTE="pasmith, post: 6528166, member: 415692"] Let [tex]\chi_A(x) = \sum_{n=0}^N a_nx^n[/tex]. Then [tex] (\chi_A(B) + \det(B)I)B^{-1} = (a_0 + \det B)B^{-1} + \sum_{n=0}^{N-1} a_{n+1}B^{n}.[/tex] Recall that [tex]B^{-1} = \frac{\operatorname{adj}(B)}{\det(B)}[/tex] where [itex]\operatorname{adj}(B)[/itex] is the adjugate matrix of [itex]B[/itex], and that [itex]a_0[/itex] vanishes as [itex]A[/itex] is singular. This shows that the limit is finite; there is more to do to show that it vanishes. EDIT: We also have the identity (stated [url=https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem#A_synthesis_of_the_first_two_proofs]here[/url]) [tex] \operatorname{adj}(-A) = \sum_{n=0}^{N-1} a_{n+1}A^n[/tex] and thus [tex] \lim_{t \to 0} (\chi_A(B) + \det(B)I)B^{-1} = \operatorname{adj}(A) + \operatorname{adj}(-A) = (1 + (-1)^{N-1})\operatorname{adj}(A)[/tex] which is either [itex]2\operatorname{adj}(A)[/itex] or [itex]0[/itex] depending on whether [itex]N[/itex] is odd or even. [/QUOTE]
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Mathematics
Linear and Abstract Algebra
Prove that the limit of this matrix expression is 0
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