**Definition:** A real square matrix $A$ is **symmetric** if it equals its transpose:

$$A = A^T \quad \text{or equivalently} \quad a_{ij} = a_{ji}$$

**Example:**
$$A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 5 & 4 \\ 3 & 4 & 6 \end{pmatrix}$$

**Properties:**
- All eigenvalues are real
- Eigenvectors corresponding to distinct eigenvalues are orthogonal
- $A$ is orthogonally diagonalizable: $A = Q\Lambda Q^T$ where $Q$ is orthogonal
- The sum and difference of symmetric matrices is symmetric
- If $A$ is invertible and symmetric, then $A^{-1}$ is symmetric

**Applications:**
- Covariance matrices in statistics
- Adjacency matrices of undirected graphs
- Hessian matrices in optimization
- Stress/strain tensors in physics

A symmetric matrix is **positive definite** if $x^T Ax > 0$ for all nonzero $x$.