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Math Help - Prove that similar matrices have the same characteristic equations

  1. #1
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    Question Prove that similar matrices have the same characteristic equations

    Hello,

    I have been asked on an assignment to prove that similar matrices have the same characteristic equations. I believe that I would like to prove that det(A-LI) = det(M^-1AM-LI). (L = lambda) The only idea I have had thus far is to take the inverse of the right hand side of that equation. The farthest I have reduced that to is det(MA^-1M^-1-LI), but even if that IS right, I don't know where to go from there. Any help would be great.

    Sorry about the exponentialized -1s. I don't know how to properly display an exponent in the math code.

    Thanks!
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  2. #2
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     x^{-1} : [tex] x^{-1} [/tex]

     \lambda : [tex] \lambda [/tex]
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  3. #3
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    Quote Originally Posted by nathron View Post
    Hello,

    I have been asked on an assignment to prove that similar matrices have the same characteristic equations. I believe that I would like to prove that det(A-LI) = det(M^-1AM-LI). (L = lambda) The only idea I have had thus far is to take the inverse of the right hand side of that equation. The farthest I have reduced that to is det(MA^-1M^-1-LI), but even if that IS right, I don't know where to go from there. Any help would be great.

    Sorry about the exponentialized -1s. I don't know how to properly display an exponent in the math code.

    Thanks!

    A\sim B\Longrightarrow A=M^{-1}BM\Longrightarrow p_A(t)=det(A-tI)=\det(M^{-1}BM-tI) =\det\left(M^{-1}(B-tI)M\right)=\det(M^{-1})\det(M)\det(B-tI)=p_B(t)

    Tonio

    Ps. Of course, justify all the steps in the above proof.
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  4. #4
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    Hey tonio,

    Thanks for the reply. When you do the step \det(M^{-1}BM-{\lambda}I) = \det\left(M^{-1}(B-{\lambda}I)M\right) , are you assuming that M^{-1}IM = I?
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    Quote Originally Posted by nathron View Post
    Hey tonio,

    Thanks for the reply. When you do the step \det(M^{-1}BM-{\lambda}I) = \det\left(M^{-1}(B-{\lambda}I)M\right) , are you assuming that M^{-1}IM = I?

    Yes, of course...and also the fact that scalar matrices commute with any matrix, so tI=tMM^{-1}= M(tI)M^{-1}...and use also left and right distributivity of matrix multiplicatio.

    Tonio
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    Thanks!

    You've been very helpful.
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