If $A$ and $B$ are $n\times n$ complex matrices , then $AB-BA=I$ is impossible
I understand this by any example. But how can one explain it generally?
If $A$ and $B$ are $n\times n$ complex matrices , then $AB-BA=I$ is impossible
I understand this by any example. But how can one explain it generally?
Another way could be to try the multiplicative identity
$$\det({\bf AB}) = \det({\bf A})\det({\bf B})$$
And consider the relation between determinant and eigenvalues
$$\det({\bf X})=\prod_{i=0}^{n} \lambda_i({\bf X})$$
together with the perturbation of eigenvalues by addition of identity:
$$\lambda_k({\bf X+I}) = 1+\lambda_k({\bf X})$$