Matrices with all zero eigenvalues

In summary, if a matrix has all eigenvalues equal to zero, then it can be put into Jordan Normal Form with zeros along the main diagonal and possibly ones on the diagonal above. However, if the matrix cannot be diagonalized, then it can still have all zero eigenvalues and be classified as a nilpotent matrix. When multiplying two matrices with all zero eigenvalues, the resulting product may not necessarily have all zero eigenvalues.
  • #1
Leo321
38
0
If I have a matrix for which all eigenvalues are zero, what can be said about its properties?
If I multiply two such matrices, will the product also have all zero eigenvalues?

Thanks
 
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  • #2
All eigenvalue of an n by n matrix, A, are 0 if and only if [itex]A^n v= 0[/itex] for all vectors, v.

If [tex]A= \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}[/tex] and [tex]B= \begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}[/tex], then A and B both have all eigenvalues 0 but AB does not.
 
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  • #3
Thanks!
How do you show that Av=0 for all vectors v?
I am not sure I understand the meaning of a matrix with all-zero eigenvalues. Obviously you can't decompose it to a diagonal representation.
 
  • #4
Leo321 said:
Thanks!
How do you show that Av=0 for all vectors v?
You don't- it isn't necessarily true. For example, take
[tex]A= \begin{bmatrix}0 & 1 \\ 0 & 0\end{bmatrix}[/tex] and
[tex]v= \begin{bmatrix}0 \\ 1\end{bmatrix}[/tex]. The only matrix A, such that "Av= 0 for all vectors v" is, of course, the 0 matrix.

What is true, as I said before, is that [itex]A^nv= 0[/itex] for all vectors v, where A is an n by n matrix.


I am not sure I understand the meaning of a matrix with all-zero eigenvalues. Obviously you can't decompose it to a diagonal representation.
Not quite obvious!
[tex]\begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}[/tex]
is such a "diagonal representation".

An n by n matrix can be "diagonalized" if and only if there exist n independent eigenvectors. If that is not true, then the matrix can be put into "Jordan Normal Form" which has its eigenvalues along the main diagonal and possibly "1"s on the diagonal above the main diagonal- with 0 elsewhere.
If A is 3 by 3 then it can be reduced to one of these three forms:
[tex]\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}[/tex]
[tex]\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}[/tex]
or
[tex]\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{bmatrix}[/tex]
depending upon whether the "eigenspace" of its eigenvectors has dimension 3, 2, or 1, respectively.
 
  • #5
Thanks
 
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  • #6
I think you are struggling with the question rather than its answer. The question says whenever I have a matrix (say n-by-n), I compute the eigenvalues with the characteristic equation and I obtain [itex]\lambda^n=0[/itex]. Then I have one more matrix as such. And when I multiply these two matrices and compute the eigenvalues, can I get the same zero eigenvalues? Actually HallsofIvy provided you such matrices for a counterexample. my suggestion is that you work out the eigenvalues of that example for both A,B and also AB.

For your original question, let me poison you with some more terminology : http://en.wikipedia.org/wiki/Nilpotent_matrix" [Broken]. You can read the properties from the link.
 
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1. What are matrices with all zero eigenvalues?

Matrices with all zero eigenvalues are square matrices that have a characteristic polynomial with all coefficients equal to zero. This means that the matrix has no non-zero eigenvalues, and therefore no distinct eigenvectors.

2. What do matrices with all zero eigenvalues represent?

Matrices with all zero eigenvalues represent transformations that scale vectors to the zero vector. This means that any vector multiplied by the matrix will result in a zero vector.

3. Can a matrix have all zero eigenvalues and still be invertible?

No, a matrix with all zero eigenvalues cannot be invertible. In order for a matrix to be invertible, it must have non-zero eigenvalues. A matrix with all zero eigenvalues is singular, meaning it does not have an inverse.

4. How do matrices with all zero eigenvalues affect the determinant?

The determinant of a matrix with all zero eigenvalues is always equal to zero. This is because the determinant is calculated by multiplying the eigenvalues, and since all the eigenvalues are zero, the product will also be zero.

5. Are there any practical applications for matrices with all zero eigenvalues?

Matrices with all zero eigenvalues have applications in data compression and machine learning. They can also be used to represent degenerate systems in physics and engineering, where the system has no distinct stable state.

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