8.610 min read

Symmetric Matrices

A square matrix $A$ is symmetric when it equals its own transpose: $A^\top = A$ . Entry-by-entry, this means $a_{ij} = a_{ji}$ — the matrix is a mirror image across its main diagonal. Familiar examples include the identity matrix and any diagonal matrix.

The key insight: every quadratic function $f(\mathbf{x}) = \mathbf{x}^\top A \mathbf{x}$ has a unique symmetric matrix representation. Given any matrix $A$ , you can replace it with $\frac{1}{2}(A + A^\top)$ without changing the quadratic form at all. This works because for a scalar $s = \mathbf{x}^\top A \mathbf{x}$ , we have $s = s^\top = \mathbf{x}^\top A^\top \mathbf{x}$ , so you can average the two representations.

The symmetric version has off-diagonal entry $(i,j)$ equal to the average of the original $a_{ij}$ and $a_{ji}$ . For example, the non-symmetric matrix $\begin{pmatrix}3 & 2 & 0\\0 & 7 & 0\\4 & 0 & 1\end{pmatrix}$ defines the same quadratic as $\begin{pmatrix}3 & 1 & 2\\1 & 7 & 0\\2 & 0 & 1\end{pmatrix}$ . From now on, we always work with symmetric matrices for quadratic forms.

Formal View

Definition 8.5 — Symmetric Matrix

n \times n

matrix

A

is symmetric if

A^\top = A

, or equivalently

a_{ij} = a_{ji}

for all

1 \leq i, j \leq n

Lemma 8.1 — Unique Symmetric Representation

For any

n \times n

matrix

A

, we have

\mathbf{x}^\top A \mathbf{x} = \mathbf{x}^\top \left(\frac{A + A^\top}{2}\right) \mathbf{x}

for all

\mathbf{x} \in \mathbb{R}^n

. The matrix

\frac{1}{2}(A + A^\top)

is symmetric and is the unique symmetric matrix representing the same quadratic form.

Without loss of generality, the matrix of a quadratic form is always assumed to be symmetric.

Interactive Visualization

Transpose Visualizer

Why This Matters

Symmetric matrices arise naturally whenever a relationship between two things is mutual — covariance, stiffness, and Hessians are all symmetric.

Covariance matrices in statistics: $\Sigma_{ij} = \text{Cov}(X_i, X_j) = \text{Cov}(X_j, X_i)$ .
Stiffness matrices in structural engineering: force-displacement relationships are mutual.
Hessian matrices of second derivatives are symmetric by Clairaut's theorem.

Learning Resources

Symmetric matrices

Khan Academy

Introduction to symmetric matrices and their properties.

7 min

Transpose of a matrix

3Blue1Brown

Geometric interpretation of the transpose operation.

4 min

Quiz

Question 1

Every diagonal matrix is symmetric.

Question 2

What is the symmetric matrix associated with the quadratic form defined by $A = \begin{pmatrix}1 & 4\\0 & 2\end{pmatrix}$ ?

Common Mistakes

Thinking symmetry is just about the diagonal entries — it is about all off-diagonal pairs: $a_{ij} = a_{ji}$ .
Forgetting to symmetrize before computing eigenvalues — technically $\mathbf{x}^\top A \mathbf{x}$ works for any $A$ , but spectral theorem requires $A$ to be symmetric.