Diagonalizing a 3x3 Matrix: Troubleshooting the P-1AP = D Expression

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Homework Help Overview

The discussion revolves around the diagonalization of a 3x3 matrix, specifically addressing the expression P-1AP = D. Participants are examining the eigenvalues and eigenvectors of the matrix and their implications for constructing the matrices P and D.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to construct the matrix P using eigenvectors but encounters issues with achieving a diagonal matrix D. Some participants question whether the columns of P were correctly formed from the eigenvectors and inquire about the calculation of P-1.

Discussion Status

Participants are actively engaging in troubleshooting the diagonalization process. Some guidance has been offered regarding the construction of matrix P, and there is an acknowledgment of potential errors in matrix multiplication. Multiple interpretations of the eigenvalues are also being explored.

Contextual Notes

There is a noted discrepancy in the eigenvalue identification, with one participant suggesting a different value for the third eigenvalue, which may influence the discussion on the matrix setup.

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I have to diagonalize this matrix and I've found the eigenvalues and vectors and they're linearly independent but I can't get the expression P-1AP = D to work.

It's a 3x3 matrix, (-11,-46,-3),(0,12,0),(-1,-2,-9) with eigenvalues of 12,-12 and [STRIKE]8[/STRIKE] -8.
The eigenvectors are (-2,1,0), (3,0,1) and (-1,0,1).

The (P) I made was using the vectors but D is never a diagonal matrix am I missing something?

Edited to correct typo.
 
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Did you make the columns of P (not the rows) out of the eigenvectors? What did you get for P-1?
 


vela said:
Did you make the columns of P (not the rows) out of the eigenvectors? What did you get for P-1?

My matrix P was (-2,3,-1), (1,0,0), (0,1,1)
 


P-1AP comes out diagonal here with that matrix.
 


IF you say it's diagonal than my matrix multiplication skills must be in need of help... I'll try again and see. Thanks for your help!
 


I get the third eigenvalue to be -8, not 8 as you showed. That wouldn't have made a difference in your calculation of P and P-1 though.
 

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