(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Diagonalization & Eigen vectors proofs

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