Thread: 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?

Not so obvious in the case .

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

Yes, the in my previous post showed.

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

It's not diagonalizable.

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?

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.