Can Any Square Matrix be Expressed as a Matrix Exponential of Some Matrix

Question

Can any square matrix, $\hat{M}$, be written as $\hat{M} = e^{i\hat{O}}$, for some square matrix $\hat{O}$? For instance, can $\hat{M} = \begin{bmatrix} \alpha & 0 &0 &0 \\0& 0 &0 &0\\0& 0 &0 &0\\0& 0 &0 &b \end{bmatrix}$? If not, what are the restrictions on $\hat{M}$, in order for the above to be possible?

Thanks!

Answer 1

No. Matrices of the form $e^A$ are always invertible (with inverse $e^{-A}$).