# Can an eigenvalue $\lambda_i=0$?

I was doing some work with diagonalization of a matrix $A$ in order to find a matrix $P$ such that $\,P^{-1}AP\,$ was diagonal. In order to that I set $\;\lambda I_{n}=0\;$ and found the characteristic polynomial and its roots.

When I factored my characteristic polynomial I obtained $\;\lambda^2(\lambda-2),\,$ so $\,\lambda=0,\,2$.

I was taught that the eignenvalues$\,\lambda_{i}\,$ I found become the entries of the diagonal matrix $\,P^{-1}AP.\,$ If this is indeed true, then two of the diagonal entries would be $\,0.\,$ Is this allowed, or must a diagonal matrix strictly have non-zero diagonal entries?

• Zero is allowed. You may be thinking of eigenvectors --- the zero vector can't be an eigenvector. But for eigenvalues, no problem. – Gerry Myerson Jun 21 '13 at 23:44

Absolutely yes: It is very possible $\lambda = 0$. Zero is allowed.

You may be mixing up what you know about eigenvectors --- the zero vector cannot be an eigenvector.

But for an eigenvalue $\lambda$, it is certainly possible and admissible that $\lambda = 0$.

With respect to your last question:

"($\lambda = 0$): Is this allowed, or does a diagonal matrix strictly have to have the diagonal entries as non-zero?"

Yes, it is allowed for zero's to be on the diagonal. No, the diagonal entries need not be non-zero.

• Dang! Beat me to the punch with the eigenvalue vs. eigenvector observation. +1 – Cameron Buie Jun 21 '13 at 23:58
• +1 what that Dang! means Amy? Dang Dang... Is that like a sound of a big Bell? – mrs Jun 22 '13 at 0:09
• +1 it's a non-profane way of saying something like "Damn": meaning, if I hadn't posted what I posted, Cameron intended to. – amWhy Jun 22 '13 at 0:12
• @amWhy Thanks; makes perfect sense now. I think I was confusing eigenvalues with eigenvectors. – Sujaan Kunalan Jun 22 '13 at 0:36
• @amWhy: Spot on +1 – Amzoti Jun 22 '13 at 0:38

A matrix is called diagonal if its off-diagonal entries ($a_{ij}$ for $i \not= j$) are all zero. This does not require the diagonal entries ($a_{jj}$) to be nonzero. For instance, the zero matrix is diagonal.