Symmetric matrices

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 $x$.