Thread: Sum of dimensions of eigenspaces?

1. 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!

2. Re: Sum of dimensions of eigenspaces?

Originally Posted by virussss123
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!
I'm confused. Can you restate the question.

3. Re: Sum of dimensions of eigenspaces?

Originally Posted by virussss123
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!
dim(span{(-2 0 1)}) + dim(span{(1 0 0)}) = 1 + 1 = 2

4. Re: Sum of dimensions of eigenspaces?

I'm trying to find out if a matrix A is diagonalisable. In order to do this, I need to work out the sum of dimensions of eigenspaces.
Does that make it clearer?

5. Re: Sum of dimensions of eigenspaces?

Originally Posted by alexmahone
dim(span{(-2 0 1)}) + dim(span{(1 0 0)}) = 1 + 1 = 2
Why does it end up as 1+1?

6. Re: Sum of dimensions of eigenspaces?

Originally Posted by virussss123
Why does it end up as 1+1?
For any non-zero vector v, dim(span(v)) = 1 because a basis of span(v) is {v}, which contains only one vector.

7. Re: Sum of dimensions of eigenspaces?

Originally Posted by alexmahone
For any vector v, dim(span(v)) = 1 because a basis of span(v) is {v}, which contains only one vector.
That makes sense, thanks. So if there are 3 vectors, would it make the sum 3? i.e if there are 3 eigenvalues resulting in 3 different eigenspaces, would the sum of dimensions of eigenspaces be 3? Thank you

8. Re: Sum of dimensions of eigenspaces?

Originally Posted by virussss123
So if there are 3 vectors, would it make the sum 3?
I guess so, as long as none of the vectors is the zero vector, in which case dim(span(v)) = 0.

9. Re: Sum of dimensions of eigenspaces?

Originally Posted by alexmahone
I guess so, as long as none of the vectors is the zero vector, in which case dim(span(v)) = 0.
That answers everything. Thank you alexmahone!