- #1
Bertrandkis
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Homework Statement
Question 1:
A) Show that if A is diagonalizable then [tex]A^{T}[/tex] is also diagonalizable.
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
Homework Statement
Question 2
If A and B are Similar matrices, then show that [tex]A^{2}[/tex] and [tex]B^{2}[/tex]
are similar
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
Homework Statement
Question 3
Every matrix A is Similar itself
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?