Invertibility and Diagonalizability

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  • #1
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Main Question or Discussion Point

Does there exist a matrix which is both not invertible and not diagonalizable? If so, please provide an example.

Thanks,
David
 

Answers and Replies

  • #2
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Is this a homework question??

What did you try alrready??
 
  • #3
Deveno
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hint: see if you can find a 2x2 matrix using only 0's and 1's with 0 determinant. i can think of 2 such matrices right off the bat that fulfil both your criteria.
 
  • #4
HallsofIvy
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One that comes to mind immediately is
[tex]\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}[/tex]

A matrix is not invertible if and only if it has 0 as an eigenvalue. A matrix is diagonalizable if the exist a basis for the space consisting of eigenvectors. Those are not contradictory.
 
  • #5
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One that comes to mind immediately is
[tex]\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}[/tex]

A matrix is not invertible if and only if it has 0 as an eigenvalue. A matrix is diagonalizable if the exist a basis for the space consisting of eigenvectors. Those are not contradictory.
Uuh, that is already a diagonal matrix...
 
  • #6
HallsofIvy
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Yes. So? A diagonal matrix is trivially "diagonalizable".

If you want a non-diagonal, diagonalizable, matrix that is not invertible, start with a diagonal matrix, say
[tex]D= \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0\end{bmatrix}[/tex]
and multiply by A and [itex]A^{-1}[/itex] where A is some invertible matrix.

For example, if
[tex]A= \begin{bmatrix}1 & 0 & 0 \\ 2 & -1 & 0 \\ 5 & -2 & 1\end{bmatrix}[/tex]
then
[tex]A^{-1}= \begin{bmatrix}1 & 0 & 0 \\ -2 & 1 & 0 \\ -1 & 2 & 1\end{bmatrix}[/tex]

and then
[tex]ADA^{-1}= \begin{bmatrix}-3 & 2 & 0 \\ -2 & -2 & 0 \\ 13 & -4 & 0\end{bmatrix}[/tex]
which is a non-invertible matrix which can be "diagonalized" to the original matrix, D.
 
  • #7
22,097
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Yes. So? A diagonal matrix is trivially "diagonalizable".
It's fine. Except that the OP want a matrix that is not diagonalizable. :tongue:
 
  • #8
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You can simply choose a nilpotent non-null matrix, as it's easy to see that the only diagonalizable AND nilpotent matrix is the null one.
for example:
01
00
 
  • #9
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How about [0 0 0]? Definitely a matrix, and definitely not invertible since it isn't even square. Also, certainly not diagonalizable.
 
  • #10
HallsofIvy
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dang! Don't you just hate those little words like "not"?

Okay, lets start with a Jordan Normal form, non-diagonal matrix:
[tex]P= \begin{bmatrix}2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 0\end{bmatrix}[/tex]
which also has 0 as an eigenvalue and so is not invertible. Using the same "A" as before,
[tex]APA^{-1}= \begin{bmatrix}0 & 1 & 0 \\ 4 & 0 & 0 \\ 8 & 1 & 0\end{bmatrix}[/tex]

That is neither invertible nor diagonalizable.
 

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