Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$. My Try: I proved that $C(M)$ is a subspace. But how...

This question in stackExchange remained unanswered. Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M_n(\mathbb F)$ what is the cardinality of $C_{M_n(\mathbb F)} (A)$ , the centralizer of $A$ in $M_n(\mathbb...

Suppose we have the $n \times n$ block matrix $$M = [M_{i,j}] = M_{1,1} \oplus \cdots \oplus M_{n,n}$$ such that each $M_{i,i}$ is also square and has exactly one eigenvalue $\lambda_i$ and $\lambda_i = \lambda_j \implies i=j$. I need to show that the only matrices which commute with $M$ are als...