This is one of those definitions that "just works" sorta like complex exponents but less meaningful. It's what computer scientists call "syntactic sugar". We have e^(x+y)=(e^x)*(e^y) then you plug it wherever you want to link addition & multiplication. Not beautiful per se.
beautiful matrix equality the determinant of a matrix exponential is equal to the exponential of the tracepic.twitter.com/RIzBqHzpJ9
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i don't quite understand what you're saying
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Given your last tweet isn’t this sort of trivial, at least for diagonal matrices? Multiplying exponentials is equivalent to just adding their exponents
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i think most proofs are trivial for diagonal matrices lol
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What does it mean to have a matrix as an exponent?
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it's defined via the taylor series
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Note e.g. M=S^(-1)D S, M^n=S^(-1)D^n S, and: e^M=S^(-1)e^D S Therefore: det(e^M)= det(S^(-1)e^D S)=det(e^D)= det diag(e^eigenvalues)= Tr(e^D)=Tr(S^(-1)e^D S)=Tr(e^M)
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det(e^M)=det (S^(-1)e^D S) =det (e^D) =Prod_i (e^λi) =(e^λ1)(e^λ2)...(e^λn) =e^(λ1+λ2+...+λn) =e^(Sum_i λi)=e^(Tr(D)) =e^(Tr(S^(-1)D S)=e^(Tr(M)) Where: D:=diag(λ1, λ2,..., λn)
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This is my favorite Matrix identity, it s magical.
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This is used a bunch in control theory!
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