Properties of Nilpotent Matrices

Question

I have $2$ questions concerning nilpotent matrices and commutativity

$\bullet$ If $M$ is an invertible matrix and $N$ is nilpotent, is then $M-N$ invertible ?

When $M$ and $N$ are commute yes, the inverse is then $(M^{k-1}+M^{k-2}N+\dots N^{k-1})M^{-k}$ if $N$ has nilpotent index $k$, but if they do not commute this is not the inverse, but there might exist one, or not ?

$\bullet$ Is the sum of $2$ nilpotent matrices again nilpotent ?

Of course under the condition that they don't commute, otherwise one can use binomial formula. I only know that the product of nilpotent matrices can be non-nilpotent

can you help ?

Answer 1

No the the first question $$ \begin{bmatrix}0&1\\1&0\end{bmatrix} - \begin{bmatrix}0&1\\0&0\end{bmatrix} = \begin{bmatrix}0&0\\1&0\end{bmatrix}.$$

No to the second question: $$ \begin{bmatrix}0&1\\0&0\end{bmatrix} + \begin{bmatrix}0&0\\1&0\end{bmatrix} = \begin{bmatrix}0&1\\1&0\end{bmatrix} $$

As you see, it is really the same example, with $$ M = \begin{bmatrix}0&1\\1&0\end{bmatrix} $$ invertible, and $$ N = \begin{bmatrix}0&1\\0&0\end{bmatrix}, \qquad M - N = \begin{bmatrix}0&0\\1&0\end{bmatrix} $$ nilpotent.

Answer 2

If M is invertible matrix of order n then the characteristic polynomial of M is $$C_M(x)=(x-a_1)^{m_1}(x-a_2)^{m_2}(x-a_3)^{m_3}... (x-a_n)^m_n$$ where $a_{i's}$ are eigenvalues and are non zero since $M$ is invertible and $\sum_{i=1}^n m_i=n$ and the characteristic polynomial of N is $$C_N(x)=x^n$$ Now the characteristic polynomial of $M-N$ is $$C_{M-N}=(x-(a_1+0))^{m_1}(x-(a_2+0))^{m_2}......(x-(a_n+0))^{m_n}$$ Thus we can see that none of the eigenvalue is zero hence $M-N$ is invertible. For the case of M and N both nilpotent the characteristic polynomial of M+N is $$C_{M+N}(x)=x^n $$ Thus all the eigenvalues are zero and thus $M+N$ is also nilpotent if both $M$ and $N$ are nilpotent