A 3x3 matrix with 1 real eigenvalue. Does there exist a non-diagonalizable 3x3 matrix that has precisely 1 real eigenvalue and a multiplicity of 1? When it comes to multiplicity I'm trying to find a matrix that would give me something like $(\lambda-1)^3$ as the eigenvalue. This factors down to $\lambda^3 - 3\lambda^2+3\lambda-1$ so you could say the multiplicity is 3 but you can also say that it only has 1 real root. So could I use this to find a non-diagonalizable 3x3 matrix with only 1 eigenvalue. So would such a matrix exist? 

Sure
\begin{bmatrix} 0&-1&0\\1&0&0\\0&0&1\end{bmatrix}
has one real eigenvalue of multiplicity 1.
