# Orthogonality on a Matrix Ring

1. " A real square matrix is orthogonal if and only if its columns form an orthonormal basis of $\mathbb{R}^n$." Im looking for a generalization of that fact in Matrix Rings.
2. If $A \in M_n(R)$ is matrix ring such that $A^{-1} = A^T$. its true that all the columns of $A$ are orthogonal?

Moreoever, i only know a few facts about Matrix Rings: Let $R$ be a commutative ring.

1. The columns of $A\in M_n(R)$ are l.i if and only if $det(A)$ is not a zero diviros.

2. The ideals of $M_n(R)$ are in bijection with the ideals of $R$. And the ideals on $M_n(R)$ are matrix with entries on a ideal $I$ in $R$.

Where i can find a book about Matrix Rings? Or a page where i can find more facts about it. Im intersted in know facts about orthogonality on a Matrix Ring.

To answer your first question, what does it mean for two vectors in $R^n$ to be orthogonal? There is a symmetric bilinear form \begin{align*} R^n \times R^n &\longrightarrow R \\ (v,w) &\longmapsto \langle v,w\rangle := \sum_{i=1}^n v_i w_i \end{align*} and you can call two vectors $v,w\in R^n$ orthogonal if $\langle v,w\rangle = 0$. If you want to do so, then yes, $A^{-1}=A^T$ is equivalent to all columns of $A$ being orthogonal. That's just the definition of the matrix product.