# Is it possible to have a diagonal matrix with all eigenvalues = zero ?

1. Mar 5, 2013

### gamerninja213

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

If the only eigenvalue is zero, can you ever get a set of n linearly independent vectors?
2. Relevant equations

3. The attempt at a solution

2. Mar 5, 2013

### jbunniii

In response to the question in the subject, the zero matrix is diagonal and all its eigenvalues are zero.

In response to the question in the problem statement, if even one eigenvalue is zero, then by definition that means Ax = 0 for some nonzero x. Thus the columns of the matrix cannot be linearly independent.

3. Mar 5, 2013

### Dick

The only eigenvalue of the zero matrix is 0. You can certainly find a set of linearly independent eigenvectors. ANY set of linearly independent vectors will do it. Is that all you are asking?

4. Mar 7, 2013

### gamerninja213

The question in the headline statement was a typo sorry.