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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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- Thread starter looth
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- #1

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Is there a theorem which classify the diagonalizable matrices??

If so, could someone please kindly tell me which journal is it.

Thanks

- #2

HallsofIvy

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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

HallsofIvy

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A sufficient condition for linear transforms in general is that the be "self-adjoint" (which reduces to being symmetric for real matrices).

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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

regards

looth

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* 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

matt grime

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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

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HallsofIvy said:

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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