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