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September 8th, 2011, 12:56 PM   #1
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Sum of dimensions of eigenspaces

How do I calculate the sum of dimensions of eigenspaces to evaluate whether or not a matrix is diagonalisable?

E.g. dim(span{(-2 0 1)}) + dim(span{(1 0 0)}) ?
They're supposed to be written as vectors but I can't work out how to do that so I apologise for that. Thank you!
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September 9th, 2011, 07:08 AM   #2
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Re: Sum of dimensions of eigenspaces

Each individual vector spans, of course, a one-dimensional subspace and the two vectors are clearly independent
(a(-2 0 1)+ b(1 0 0)= (-2a+ b 0 a)= (0 0 0) if and only if -2a+ b= 0 and a= 0. From a= 0, b= 0.)
and so together they span a two-dimensional subspace.

If I were you I would not worry about "dimension" but about showing that the vectors are independent. If a set of n vectors are independent, then they span an n-dimensional space. If, in a set of n vectors, the largest subset of independent vectors contains m vectors, then they span a subspace of dimension m.
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