# Invertible Matrix corresponds to Automorphism

Jump to navigation
Jump to search

This article needs to be linked to other articles.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

## Theorem

Let $R$ be a ring with unity.

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Let $G$ be an $n$-dimensional $R$-module.

Let $\map {\MM_R} n$ be the $n \times n$ matrix space over $R$.

Let $\map {\LL_R} G$ be the set of all linear operators on $G$.

Then the invertible elements of the ring of square matrices $\struct {\map {\MM_R} n, +, \times}$ correspond directly to automorphisms of $\map {\LL_R} G$.

## Proof

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices