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Diagonalization & Eigen vectors proofs 
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#1
Jan1008, 06:22 AM

P: 25

1. The problem statement, all variables and given/known data
Question 1: A) Show that if A is diagonalizable then [tex]A^{T}[/tex] is also diagonalizable. 3. The attempt at a solution We know that [tex]A[/tex] is diagonalizable if it's similar to a diagonal matrix. So [tex]A[/tex]=[tex]PDP^{1}[/tex] [tex]A^{T}[/tex]=[tex](PDP^{1})^{T}[/tex] which gives [tex]A^{T}[/tex]=[tex](P^{1})^{T}DP^{T}[/tex] as [tex]D=D^{T}[/tex] Hence [tex]A^{T}[/tex] is diagonalizable 1. The problem statement, all variables and given/known data Question 2 If A and B are Similar matrices, then show that [tex]A^{2}[/tex] and [tex]B^{2}[/tex] are similar 3. The attempt at a solution If A and B are similar then [tex]P^{1}AP[/tex] = [tex]B[/tex] We know that [tex]P^{1}A^{k}P[/tex] =[tex]D^{k}[/tex] let k=2 therefore [tex]P^{1}A^{2}P[/tex] =[tex]B^{2}[/tex] hence [tex]A^{2}[/tex] and [tex]B^{2}[/tex] are similar 1. The problem statement, all variables and given/known data Question 3 Every matrix A is Similar itself 3. The attempt at a solution If A and A are similar then [tex]P^{1}AP[/tex] =[tex]A[/tex] ? this does not make sense to me. Alternatively, do we have to show that A has the same eigenvalues as A? This is obvious, is this then the proof? 


#2
Jan1008, 06:27 AM

Math
Emeritus
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Thanks
PF Gold
P: 39,344

B^{2}= (P^{1}AP)(P^{1}AP)= (P^{1}A)(PP^{1})(AP). 


#3
Jan1408, 02:12 AM

P: 25

Thanks for the reply, I see where I went wrong.
I tried to use the method in question 2 and extend it to prove that : IF A and B are similar matrices then [tex]A^{k}[/tex] and [tex]B^{k}[/tex] are similar for any non negative integer k. This is what I got: [tex]B^{k}[/tex]=[tex](P^{1}AP)[/tex] [tex](P^{1}AP)[/tex] ......[tex](P^{1}AP)[/tex] (k times) then Multiply the right hand side 2 elements at a time as u did we will end up with [tex]P^{1}A^{k}P[/tex]. Is This the correct way to proove it? 


#4
Jan1408, 05:22 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,344

Diagonalization & Eigen vectors proofs
Yes, that works nicely.



#5
Jan1408, 04:06 PM

P: 182




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