Classification of diagonalizable matrices

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Discussion Overview

The discussion revolves around the classification and characterization of diagonalizable matrices, exploring both necessary and sufficient conditions for various subclasses of these matrices. Participants engage in theoretical reasoning and clarification of concepts related to diagonalizability, normal matrices, and eigenvectors.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that a matrix is diagonalizable if there exists a complete set of eigenvectors, which forms a basis for the vector space.
  • Others propose that normal matrices are diagonalizable, and inquire about conditions for non-normal diagonalizable matrices.
  • A participant mentions that a sufficient condition for real matrices to be diagonalizable is symmetry, while self-adjoint matrices are noted as a special case of normal matrices.
  • Criteria for diagonalizability are discussed, including the requirement that eigenvectors form a basis and that algebraic multiplicities equal geometric multiplicities.
  • One participant emphasizes that the characteristic polynomial having no multiple roots is not equivalent to the other criteria, which are considered equivalent.
  • Another participant asserts that the only true characterization for an n x n matrix is to have n eigenvectors, suggesting a bijection with C^n under certain conditions.
  • There is a request for a formal theorem that outlines necessary conditions for diagonalizability, particularly in relation to normal matrices.

Areas of Agreement / Disagreement

Participants express differing views on the characterization of diagonalizable matrices, with no consensus reached on a definitive classification or necessary conditions. Multiple competing perspectives on the criteria for diagonalizability remain present.

Contextual Notes

Some criteria mentioned are not equivalent, and the discussion highlights the complexity of establishing necessary and sufficient conditions for different subclasses of diagonalizable matrices. The relationship between normal and non-normal matrices is also a point of contention.

looth
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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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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.
 
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
 
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).
 
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:
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
 
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...
 
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
 

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