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

It seems obvious, but how to prove it?

Printable View

- November 24th 2012, 01:06 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? - November 24th 2012, 02:51 AMgirdavRe: Rank one matrix has a nonzero eigenvalue?
Not so obvious in the case .

- November 24th 2012, 03:07 AMwesleybrownRe: Rank one matrix has a nonzero eigenvalue?
Oops! Then, is it true that not all rank one matrices are diagonalizable?

- November 24th 2012, 03:14 AMgirdavRe: Rank one matrix has a nonzero eigenvalue?
Yes, the in my previous post showed.

- November 24th 2012, 03:19 AMwesleybrownRe: Rank one matrix has a nonzero eigenvalue?
If it is rank one and symmetric matrix, how do we show it is diagonalizable?

- November 24th 2012, 03:23 AMgirdavRe: Rank one matrix has a nonzero eigenvalue?
It's

*not*diagonalizable. - November 24th 2012, 03:31 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? - November 24th 2012, 04:14 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.

- September 4th 2014, 07:40 PMphys251Re: Rank one matrix has a nonzero eigenvalue?
- September 5th 2014, 05:46 AMHallsofIvyRe: Rank one matrix has a nonzero eigenvalue?
Yes, that was the

**point**of girdav's example. - September 5th 2014, 09:20 AMphys251Re: Rank one matrix has a nonzero eigenvalue?