A matrix invariant For a square matrix $M$ and a nonzero natural integer $k$, we denote $$\delta_k(M) = -\operatorname{dim}\ker M^{k-1} + 2\operatorname{dim}\ker M^k - \operatorname{dim}\ker M^{k+1}.$$ a) Prove that if two square matrices $M$ and $M'$ are similar, then $\delta_k(M) = \delta_k(M')$ for all $k$. b) Let $r$ be a nonzero natural integer. Verify that for all nonzero integer $k$, $\delta_k(J_r)$ equals 1 if $k = r$ and 0 otherwise. c) Let $M_1$ and $M_2$ be two square matrices and let $M = \operatorname{diag}(M_1, M_2)$. Prove the relation $\operatorname{dim}\ker M = \operatorname{dim}\ker M_1 + \operatorname{dim}\ker M_2$ and then that for all nonzero integer $k$, $$\delta_k(M) = \delta_k(M_1) + \delta_k(M_2).$$ You may use without proof the fact that all these relations extend to a block diagonal matrix $\operatorname{diag}(M_1, \ldots, M_s)$.
\textbf{A matrix invariant}\\
For a square matrix $M$ and a nonzero natural integer $k$, we denote
$$\delta_k(M) = -\operatorname{dim}\ker M^{k-1} + 2\operatorname{dim}\ker M^k - \operatorname{dim}\ker M^{k+1}.$$
a) Prove that if two square matrices $M$ and $M'$ are similar, then $\delta_k(M) = \delta_k(M')$ for all $k$.
b) Let $r$ be a nonzero natural integer. Verify that for all nonzero integer $k$, $\delta_k(J_r)$ equals 1 if $k = r$ and 0 otherwise.
c) Let $M_1$ and $M_2$ be two square matrices and let $M = \operatorname{diag}(M_1, M_2)$. Prove the relation $\operatorname{dim}\ker M = \operatorname{dim}\ker M_1 + \operatorname{dim}\ker M_2$ and then that for all nonzero integer $k$,
$$\delta_k(M) = \delta_k(M_1) + \delta_k(M_2).$$
You may use without proof the fact that all these relations extend to a block diagonal matrix $\operatorname{diag}(M_1, \ldots, M_s)$.