Question concerning Aldrovandi/Freitas-article

[Gottfried]

Hi -

maybe this was already discussed here, but rereading Aldrovandi/Freitas I find a remark, which seems to contradict my diagonalization in the exp(x)-1-iteration.

They state pg 16, concerning the triangular Bell-matrix, (U or S2 in my notation)

"(...) Bell matrices are not normal, that is, they do not commute with their transposes. Normality is the condition for diagonalizability. This means that Bell matrices cannot be put into diagonal form by a similarity transformation. (...)"

In my understanding this remark is a bit misleading; the normality-criterion applies only, if an orthonormal similarity transform is requested, which is usually also called a rotation. But here we are able to do a similarity transform using triangular matrices, which even allows exact powerseries-terms for arbitrary size of matrices.

Did I overlook something?

Gottfried
R. Aldrovandi and L.P.Freitas; Continuous iteration of dynamical maps; 1997; Online at arXiv physics/9712026 16.dec 1997

wikipedia:diagonalizable
wikipedia:normal matrix

[bo198214]

"(...) Bell matrices are not normal, that is, they do not commute with their transposes. Normality is the condition for diagonalizability. This means that Bell matrices cannot be put into diagonal form by a similarity transformation. (...)"

In my understanding this remark is a bit misleading; the normality-criterion applies only, if an orthonormal similarity transform is requested, which is usually also called a rotation. But here we are able to do a similarity transform using triangular matrices, which even allows exact powerseries-terms for arbitrary size of matrices.

Dont understand this either. I think they are wrong.

[andydude]

So what is the difference between "normality" and "distinct eigenvalues"? I thought that distinct eigenvalues were sufficient for diagonizability...

Andrew Robbins

[bo198214]

So what is the difference between "normality" and "distinct eigenvalues"? I thought that distinct eigenvalues were sufficient for diagonizability...

yes, but not for normality.
We have the implications:

normal -> diagonizable
distinct eigenvalues -> diagonizable

but not the reverse directions.

[Gottfried]

So what is the difference between "normality" and "distinct eigenvalues"? I thought that distinct eigenvalues were sufficient for diagonizability...

Andrew Robbins

"normal": let M be a matrix (we're discussing real matrices for example). Then M is "normal", if M commutes with its transpose M*M' = M'*M
This equality is obviously true for symmetric M, but also for some others.

It is said, that for normal matrices, if
T*M*T^-1 = D , D diagonal,
then T is orthogonal, meaning T*T' = T*T^-1=I (I think T is always a rotation)
and also

T*M*T' = D

(from other context I'm used to denote rotation-matrices by letter T)
---------------------------

not "normal", but still diagonalizable (the more general case):

W*M*W^-1 = D

no specific properties on W.

Related to current discussion: if M is triangular (and diagonalizable), I think W is also triangular (but I must check this), and the eigenvalues are the entries of its diagonal.

-------------------

The question whether eigenvalues are distinct or not is not relevant here; this is only relevant for the description of further properties of W (whether it is unique ... )

Gottfried