Symmetric Matrix¶

Definition¶

A symmetric matrix is a real square matrix $\boldsymbol{A}$ satisfying $\boldsymbol{A}=\boldsymbol{A}^{\top}$ $A^{⊤}$ . Equivalently, its entries satisfy $a_{ij}=a_{ji}$ for all $i,j$ .

Interpretation¶

The transpose exchanges entry $a_{ij}$ with $a_{ji}$ . Equality with the transpose therefore requires matching entries on the two sides of the diagonal, not merely matching diagonal entries.

A symmetric matrix is unchanged by reflection across its main diagonal.

A 3 by 3 matrix symmetric about its main diagonal, next to the two eigenvectors of a 2 by 2 symmetric matrix, which meet at a right angle.

The eigenvectors $\boldsymbol{p}_1,\boldsymbol{p}_2$ of the symmetric matrix $\begin{pmatrix}2&1\\1&2\end{pmatrix}$ are orthogonal. In general a real symmetric matrix has real eigenvalues and an orthonormal eigenbasis: $\boldsymbol{A}=\boldsymbol{Q}\boldsymbol{\Lambda} \boldsymbol{Q}^{\top}$ where $\boldsymbol{\Lambda}$ is the diagonal matrix of eigenvalues and $\boldsymbol{Q}$ is orthogonal, $\boldsymbol{Q}^{\top}\boldsymbol{Q}=\boldsymbol{I}$ $I$ , with mutually orthogonal unit eigenvectors as columns (spectral theorem).

With $\boldsymbol{A}=\boldsymbol{A}^{\top}$ the eigenvector lines are perpendicular. Switch the symmetry off and change $a_{21}$ : the right angle is lost.

Key Results¶

Every real diagonal matrix is symmetric.
If $\boldsymbol{A}$ and $\boldsymbol{B}$ are symmetric, then $\boldsymbol{A}+\boldsymbol{B}$ is symmetric, but $\boldsymbol{A}\boldsymbol{B}$ need not be symmetric.

Examples¶

$\begin{pmatrix}1&2\\2&5\end{pmatrix}$ is symmetric.
$\begin{pmatrix}1&2\\3&4\end{pmatrix}$ is not symmetric.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 3

Aliases: self-adjoint matrix Symbols: lambda

Last updated: 20 October 2026 03:02