My Math Forum Sum of dimensions of eigenspaces

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 September 8th, 2011, 12:56 PM #1 Newbie   Joined: Sep 2011 Posts: 6 Thanks: 0 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!
 September 9th, 2011, 07:08 AM #2 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 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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