Example of $2\times2$ matrix with singular values $0$ and $5$ but only has one eigenvalue $0$. My understanding of singular values is that they are the square roots of eigenvalues, but I am definitely missing something here in the definition. 
The problem I am trying to work on is: 
Find an example of $T \in L(\mathbb{C}^2)$ such that $0$ is the only eigenvalue of $T$
and the singular values of $T$ are $5$, $0$. 

The singular values of $A$ are the square roots of the eigenvalues of $A^*A$ (or of $AA^*$), not those of $A$ itself. For one thing, $A$ could have negative eigenvalues.
For your specific problem, try $A=\begin{bmatrix}0&5\\0&0\end{bmatrix}$.
