# Homework Help: Null space and eigenspace of diagonal matrix

1. Nov 1, 2012

1. The problem statement, all variables and given/known data

I am working on a problem where I made a matrix representation of a linear transformation and I am asked what is the eigenspace for a particular eigenvalue.

2. Relevant equations

3. The attempt at a solution
The problem for me is, I came out with a diagonal matrix. what is the null space of a diagonal matrix? Is it just the 0 vector? If so, can I have an eigenspace?

2. Nov 1, 2012

### Dick

Suppose your diagonal matrix is 3x3. Think about what happens when you multiply your matrix by the vectors [1,0,0], [0,1,0] and [0,0,1]. Aren't those all eigenvectors? What are their eigenvalues in terms of the diagonal matrix entries? A null space is just the set of vectors that have zero eigenvalue, right?

3. Nov 1, 2012

### hedipaldi

The eigenvalues of a diagonal matrix are on the diagonal.The eigenspace of an eigenvalue a is the nullspace of A-aI,that is,the solutions of (A-aI)x=0

4. Nov 1, 2012

### HallsofIvy

What do you mean by "came out with" a diagonal matrix? IF the matrix you are dealing with is diagonal, then its eigenvalues are the numbers on the diagonal as Hedipaldi said. An if is the diagonal number in the ith row, its eigenspace is spanned by the ith column. (If the same eigenvalue appears k times on the diagonal, the eigenspace is the space spanned by those k columns.)

But is "came out with" a diagonal matrix means you derived a diagonal matrix somehow from the given matrix we would have to know how it was derived.

5. Nov 1, 2012