interior of diagonalizable matrices

alekk

what is the interior in Mn(C) of the diagonalizable matrices ? and in Mn(R)

**Posted:** Tue Nov 02, 2004 4:13 am

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Please, could you rephrase your question in the form of the answer?

grobber

In the first case (when we deal with complex matrices), I believe the interior consists of those diagonalizable matrices which have distinct eigenvalues. Let $\mathcal D$ be the set of complex diagonalizable matrices.

We may as well assume that we're working with diagonal matrices, and if there are two equal eigenvalues, then the first two diagonal entries are equal.

If the matrix has distinct eigenvalues, then all the matrices in a sufficiently small ball around our matrix have distinct eigenvalues, because the characteristic polynomial, which is $\det(xI_n-A)$ , depends continuously on $A$ .

On the other hand, if the matrix $A$ has the first two diagonal entries equal, then we can take $A_k\to A$ , where $A_k$ is exactly the same as $A$ , except that it has a $\frac 1k$ in position $(2,1)$ . It's easy to see that this sort of matrix can be brought through a similarity to a non-diagonal Jordan form (which is the same as $A_k$ , except that instead of that $\frac 1k$ it has a $1$ ), so it's not diagonalizable. This means that $A$ is a limit point of the complement of $\mathcal D$ .

**Posted:** Sat May 07, 2005 4:19 am

alekk

nice.

and for $Mn_(R)$ ? It is a little harder.

**Posted:** Sat May 07, 2005 8:15 am

_________________
Please, could you rephrase your question in the form of the answer?

grobber

Should the answer be the same? That's what I'm getting.

For diagonaliuzable matrices with two equal eigenvalues, we can do the exact same thing as above to prove that they are not interior points. As for the ones with distinct eigenvalues, the matrices in a ball of small enough radius centered in such a matrix will have distinct eigenvalues. The problem is that they might be complex. However, they are also roots of a real polynomial, so if there are any complex eigenvalues, they come in pairs of conjugates. If the ball is small enough, the real parts of the eigenvalues of the matrices inside the ball will also be distinct, because this is the case for our matrix, so these eigenvalues must be real.

Is it Ok? Confused

**Posted:** Sat May 07, 2005 9:13 am