# Math Help - Rank one matrix has a nonzero eigenvalue?

1. ## Rank one matrix has a nonzero eigenvalue?

How can I show that a rank one matrix has a nonzero eigenvalue?
It seems obvious, but how to prove it?

2. ## Re: Rank one matrix has a nonzero eigenvalue?

Not so obvious in the case $A:=\begin{pmatrix}0&1\\0&0\end{pmatrix}$.

3. ## Re: Rank one matrix has a nonzero eigenvalue?

Oops! Then, is it true that not all rank one matrices are diagonalizable?

4. ## Re: Rank one matrix has a nonzero eigenvalue?

Yes, the $A$ in my previous post showed.

5. ## Re: Rank one matrix has a nonzero eigenvalue?

If it is rank one and symmetric matrix, how do we show it is diagonalizable?

6. ## Re: Rank one matrix has a nonzero eigenvalue?

It's not diagonalizable.

7. ## Re: Rank one matrix has a nonzero eigenvalue?

Maybe my expression is too poor.
I am talking about another matrix now, that is having rank one and it is symmetric, with these two criteria, it should be diagonalizable, and how to prove it?

8. ## Re: Rank one matrix has a nonzero eigenvalue?

Sorry, it's not your expression but my reading which is poor. A symmetric matrix with real entries is diagonalizable (it's a general result, known as spectral theorem, and doesn't use the fact that the rank is 1), and the involved diagonal matrix has rank 1. So there is in this case a non-zero eigenvalue.

9. ## Re: Rank one matrix has a nonzero eigenvalue?

Originally Posted by girdav
Not so obvious in the case $A:=\begin{pmatrix}0&1\\0&0\end{pmatrix}$.
Both eigenvalues of this matrix are 0.

10. ## Re: Rank one matrix has a nonzero eigenvalue?

Yes, that was the point of girdav's example.

11. ## Re: Rank one matrix has a nonzero eigenvalue?

Originally Posted by HallsofIvy
Yes, that was the point of girdav's example.
Ah!