andydude Wrote: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)

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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.

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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

Gottfried Helms, Kassel