How can I show that a rank one matrix has a nonzero eigenvalue?

It seems obvious, but how to prove it?

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- Nov 24th 2012, 12:06 AMwesleybrownRank 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? - Nov 24th 2012, 01:51 AMgirdavRe: Rank one matrix has a nonzero eigenvalue?
Not so obvious in the case $\displaystyle A:=\begin{pmatrix}0&1\\0&0\end{pmatrix}$.

- Nov 24th 2012, 02:07 AMwesleybrownRe: Rank one matrix has a nonzero eigenvalue?
Oops! Then, is it true that not all rank one matrices are diagonalizable?

- Nov 24th 2012, 02:14 AMgirdavRe: Rank one matrix has a nonzero eigenvalue?
Yes, the $\displaystyle A$ in my previous post showed.

- Nov 24th 2012, 02:19 AMwesleybrownRe: Rank one matrix has a nonzero eigenvalue?
If it is rank one and symmetric matrix, how do we show it is diagonalizable?

- Nov 24th 2012, 02:23 AMgirdavRe: Rank one matrix has a nonzero eigenvalue?
It's

*not*diagonalizable. - Nov 24th 2012, 02:31 AMwesleybrownRe: 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? - Nov 24th 2012, 03:14 AMgirdavRe: 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.

- Sep 4th 2014, 06:40 PMphys251Re: Rank one matrix has a nonzero eigenvalue?
- Sep 5th 2014, 04:46 AMHallsofIvyRe: Rank one matrix has a nonzero eigenvalue?
Yes, that was the

**point**of girdav's example. - Sep 5th 2014, 08:20 AMphys251Re: Rank one matrix has a nonzero eigenvalue?