Diagonalizable Matrix Proof

  • Thread starter trojansc82
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  • #1
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Homework Statement



Prove that if the eigenvalues of a diagonalizable matrix are all + or -1, then the matrix is equal to its inverse.

i) Let D = P-1AP, where D is a diagonal matrix with + or -1 along its main diagonal.

ii) Find A in terms of P, P-1, and D.

iii) Use the fact that D is the diagonal and the properties of the inverse of a product of matrices to expand to find A-1.

iv) Conclude that A-1 = A.



Homework Equations





The Attempt at a Solution



D * P-1 = P-1 AP *P-1

P * D * P-1 = P * P-1 A

PDP-1 = A

Not sure if I'm heading in the right direction. I am drawing a blank here.
 

Answers and Replies

  • #2
HallsofIvy
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Yes, [itex]PDP^{-1}= A[/itex]. Further more if D is invertible (and it clearly is since it does not have a 0 on its diagonal), so is A and [itex]A^{-1}= (PDP^{-1})^{-1}[/itex]

Now use the fact that [itex](ABC)^{-1}= C^{-1}B^{-1}A^{-1}[/itex].
 
  • #3
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Ok, how do I show that D-1 = D ?
 
  • #4
HallsofIvy
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What is the inverse of
[tex]\begin{bmatrix}a & 0 \\ 0 & b\end{bmatrix}[/tex]

What is the inverse of
[tex]\begin{bmatrix}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{bmatrix}[/tex]
 
  • #5
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1. 1/ab * [tex]\begin{bmatrix}b & 0 \\ 0 & a\end{bmatrix}[/tex]

2. 1/abc * [tex]\begin{bmatrix}c & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & a\end{bmatrix}[/tex]
 

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