Matrix Centralizers and their Applications

The talk is based on the works [1, 2, 3].

For a matrix $A\in M_n (\mathbb F)$ its centralizer

$$ \mathcal C (A)=\{X\in M_n (\mathbb F) | AX=XA \} $$
is the set of all matrices commuting with $A$.

For a set $S \subseteq M_n (\mathbb F)$ its centralizer

$$ \mathcal C (S)=\{X\in M_n (\mathbb F) | AX=XA \text{ for every } A \in S\}=\bigcap_{A \in S} \mathcal C(A) $$
is the intersection of centralizers of all its elements. Centralizers are important and useful both in fundamental and applied sciences.

A non-scalar matrix $A\in M_n (\mathbb F)$ is minimal if for every $X\in M_n (\mathbb F)$ with $\mathcal C(A) \supseteq \mathcal C (X)$ it follows that $\mathcal C(A)=\mathcal C(X)$. A non-scalar matrix $A\in M_n (\mathbb F)$ is maximal if for every non-scalar $X\in M_n (\mathbb F)$ with $\mathcal C(A) \subseteq \mathcal C (X)$ it follows that $\mathcal C(A)=\mathcal C(X)$.

We investigate and characterize minimal and maximal matrices over arbitrary fields.

Our results are then applied to the theory of commuting graphs of matrix rings and to characterize commutativity preserving maps on matrices.

[1] G. Dolinar, A.E. Guterman, B. Kuzma, P. Oblak, Commuting graphs and extremal centralizers, Ars Mathematica Contemporanea, 7(2), 2014, 453-459.

[2] G. Dolinar, A.E. Guterman, B. Kuzma, P. Oblak, Commutativity preservers via matrix centralizers, Publicationes Mathematicae Debrecen, 84(3-4), 2014, 439–450.

[3] G. Dolinar, A.E. Guterman, B. Kuzma, P. Oblak, Extremal matrix centralizers, Linear Algebra and its Applications, 438(7), 2013, 2904-2910.