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

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In summary, the conversation discusses the process of diagonalizing a 3x3 matrix and finding its eigenvalues and eigenvectors. The speaker has found the eigenvalues and vectors to be linearly independent, but is struggling to get the expression P-1AP = D to work. They mention using the eigenvectors to create P, but are unsure of the correct process. After some discussion and clarification, the speaker realizes their mistake and plans to try again. A correction is also made regarding the third eigenvalue.
  • #1
F1fan
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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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  • #2


Did you make the columns of P (not the rows) out of the eigenvectors? What did you get for P-1?
 
  • #3


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)
 
  • #4


P-1AP comes out diagonal here with that matrix.
 
  • #5


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!
 
  • #6


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.
 

1. What does it mean to "diagonalize" a matrix?

Diagonalizing a matrix involves finding a new matrix that is similar to the original matrix, but with all of its off-diagonal elements equal to zero. This new matrix is called a diagonal matrix, and it simplifies many mathematical calculations involving the original matrix.

2. When should I diagonalize a matrix?

Diagonalizing a matrix is particularly useful when performing calculations involving powers of the matrix, such as finding the matrix exponential or solving systems of linear differential equations. It can also provide insight into the properties of the original matrix.

3. How do I diagonalize a matrix?

The process of diagonalizing a matrix involves finding the eigenvalues and eigenvectors of the matrix. The eigenvectors form the columns of a matrix, and the eigenvalues form the diagonal elements of a diagonal matrix. This diagonal matrix is then multiplied by the inverse of the matrix of eigenvectors to obtain the diagonalized matrix.

4. Can any matrix be diagonalized?

Not all matrices can be diagonalized. For a matrix to be diagonalizable, it must have a complete set of linearly independent eigenvectors. This means that the matrix must have as many distinct eigenvalues as its dimension.

5. What are the benefits of diagonalizing a matrix?

Diagonalizing a matrix can greatly simplify mathematical calculations involving the matrix. It can also reveal important properties of the matrix, such as its eigenvalues and eigenvectors. Furthermore, diagonal matrices are easy to manipulate and can provide insight into the structure of a matrix.

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