Classification of diagonalizable matrices

In summary, a matrix is diagonalizable if and only if there exists a "complete set of eigenvectors". Self-adjoint matrix is a special type of normal matrix.
  • #1
looth
5
0
Hi,
Is there a theorem which classify the diagonalizable matrices??
If so, could someone please kindly tell me which journal is it.
Thanks
 
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  • #2
I'm not sure you would call it a "classification" but a matrix is diagonalizable if and only if there exist a "complete set of eigenvectors". That is, if there exist a basis for the vector space consisting of eigenvectors of the matrix.
 
  • #3
HallsofIvy said:
there exist a basis for the vector space consisting of eigenvectors of the matrix.

Thanks for your reply. For me, that's so called characterization.
Maybe I should put it in this way. Normal matrices are diagonalizable nand there also exist some non-normal, diagonalizable matrices. My question is: Are there any necessarly and sufficent conditions that characterize those subclasses of matrices??

thanks & regards
looth
 
  • #4
A sufficient condition for real matrices is that they be symmetric.

A sufficient condition for linear transforms in general is that the be "self-adjoint" (which reduces to being symmetric for real matrices).
 
  • #5
Self-adjoint matrix is a special type of normal matrix. What I'm interested is the characterization for those non-normal, diogonalizable matrices.

regards
looth
 
Last edited:
  • #6
I'm still not sure what you mean, but these are some criteria I know of:
* the eigenvectors are a basis for the vectorspace
* all the algebraic multplicities equal the geometrical multiplicities (the vectorspace is a direct sum of its eigenspaces)
* the characteristic polynom has no multiple roots
 
  • #7
It should be pointed out that those are not equivalent criteria (the last one is not implied by either of the first two, which are equivalent).

The only true characterization is (for an nxn matric) to have n eigen vectors. That is it, exactly. Nothing more, nothing less. These matrices are in bijection with C^n (assuming we're working over C here) modulo the relation x~y iff there is a permutation of the coordinates of x that gives y. That might be an interesting space to look at...
 
  • #8
HallsofIvy said:
I'm not sure you would call it a "classification" but a matrix is diagonalizable if and only if there exist a "complete set of eigenvectors". That is, if there exist a basis for the vector space consisting of eigenvectors of the matrix.

What I mean is a theorem like this:

Let M be a diagonalizable matrix. Then precisely (or maybe either) one of the following holds:
1. M is normal.
2. ...
3. ...
etc.

I only manage to state one of the case because
so far the only "nice" necessary condition (for a matrix to be diagonalizable) that I know is being normal matrix.

Regards
looth
 

1. What is a diagonalizable matrix?

A diagonalizable matrix is a square matrix that can be transformed into a diagonal matrix through a similarity transformation. This means that it has a full set of linearly independent eigenvectors and can be expressed as a product of its eigenvectors and eigenvalues.

2. How do you determine if a matrix is diagonalizable?

A matrix is diagonalizable if it has a full set of linearly independent eigenvectors. This can be determined by calculating the eigenvalues and eigenvectors of the matrix and checking for linear independence. If there are as many linearly independent eigenvectors as the size of the matrix, then it is diagonalizable.

3. What are the benefits of diagonalizable matrices?

Diagonalizable matrices have many benefits in applications. They are easier to work with because they are in diagonal form, making calculations and operations simpler. They also have unique properties that make them useful in solving linear systems of equations and in finding the behavior of dynamical systems.

4. Can a non-square matrix be diagonalizable?

No, a non-square matrix cannot be diagonalizable because in order for a matrix to be diagonalizable, it must have an equal number of rows and columns. However, a rectangular matrix can have a diagonalizable square submatrix.

5. How is the diagonalization of a matrix related to eigenvalues and eigenvectors?

The diagonalization of a matrix is directly related to its eigenvalues and eigenvectors. The eigenvectors of a matrix are used to form the diagonalization matrix, and the eigenvalues are used to form the diagonal matrix. This process allows us to express the original matrix in terms of its eigenvalues and eigenvectors, making it easier to work with.

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