44. Symmetric Matrix

A square matrix that equals its own transpose:

Symmetric matrices are mirror-images across the main diagonal.

44.1. Skew-symmetric matrices

The mirror flips sign: ${𝐴}^{𝑇}=-𝐴$, equivalently ${𝐴}_{𝑖𝑗}=-{𝐴}_{𝑗𝑖}$.

Diagonal entries must be zero (since ${𝐴}_{𝑖𝑖}=-{𝐴}_{𝑖𝑖}$).

44.2. Why symmetric matrices are special

• Real eigenvalues: every eigenvalue of a real symmetric matrix is real
• Orthogonal eigenvectors: eigenvectors corresponding to distinct eigenvalues are orthogonal

• Spectral theorem: every real symmetric matrix is orthogonally diagonalizable:

  $$𝐴=𝑄𝐷{𝑄}^{𝑇}$$

  where $𝑄$ is orthogonal and $𝐷$ is diagonal of eigenvalues

• Defines a quadratic form ${\overrightarrow{𝑥}}^{𝑇}𝐴\overrightarrow{𝑥}$

44.3. Decomposing any matrix

Any square matrix $𝑀$ uniquely splits into a symmetric and skew-symmetric part:

44.4. Where they show up

• Covariance matrices — always symmetric (and positive semi-definite)
• Quadratic forms — only the symmetric part matters
• Hessian matrices — symmetric for ${𝐶}^2$ functions (Schwarz’s theorem)
• Gram matrices ${𝑋}^{𝑇}𝑋$ — always symmetric and positive semi-definite
• Adjacency matrices of undirected graphs — symmetric