Symmetric Matrix¶

Definition¶

A symmetric matrix is a real square matrix $\boldsymbol{A}$ satisfying $\boldsymbol{A}=\boldsymbol{A}^{\top}$${\mathbf{A}}^{\top }$. 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.

Read the left panel by pairing entries in positions $(i,j)$ and $(j,i)$. Each pair must match; diagonal entries can have any real values. For instance, changing only $a_{12}$ while keeping $a_{21}$ fixed generally breaks symmetry. This entry-by-entry test is all that is needed here.

Later connection: eigenvectors (Week 6)¶

The right panel and the optional interaction below anticipate eigenvectors from Week 6. They are a consequence to revisit after learning that concept, rather than a test for understanding the present definition.

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}$$\mathbf{I}$, with mutually orthogonal unit eigenvectors as columns (spectral theorem).

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¶

• Matrix
• Transpose

Related Concepts¶

• Quadratic Form
• Diagonalization
• Eigenvalue and Eigenvector
• Orthonormal Basis

Sources¶

• Math II: Linear Algebra · Week 3

Aliases: self-adjoint matrix Symbols: lambda

Last updated: 27 October 2026 11:52