Symmetric matrix

Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $n \times n$ square matrix over $R$. The matrix $A$ is symmetric when $A^T = A$, where $A^T$ is the transpose of $A$. Equivalently, $a_{ij} = a_{ji}$ for all $i$ and $j$.

Remarks

The matrix $\begin{pmatrix} 1 & 7 \\ 7 & 2 \end{pmatrix}$ is symmetric. For each $m \times n$ matrix $B$ over a commutative ring, $B^T B$ and $B B^T$ are symmetric. A real symmetric matrix has real eigenvalues and an orthonormal basis of eigenvectors, by the spectral theorem. Symmetric matrices represent symmetric bilinear forms in a basis.

Explanations

L1 human

A symmetric matrix looks the same after we flip it over its main diagonal. The entry in row $i$, column $j$ equals the entry in row $j$, column $i$. Distance tables between cities are symmetric, because the distance from $A$ to $B$ equals the distance from $B$ to $A$.