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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 onedimensional 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 twodimensional 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 ndimensional 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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