Symmetric Matrix

• categories: Linear algebra, Matrix

Definition
A Matrix $A\in {\mathbb{R}}^{n\times n}$ is called a symmetric matrix if $A=A^{\top }$, where $A^{\top }$ denotes the transpose of $A$. Formally:
$a_{ij}=a_{ji}\forall i,j\in \{1,\dots ,n\}.$

Intuition
A symmetric matrix is one where the entries are mirrored across the diagonal, meaning the upper triangular portion is a reflection of the lower triangular portion.

Key Properties

1. Real Eigenvalues: All eigenvalues of a symmetric matrix are real.
2. Orthogonal Eigenvectors: The eigenvectors corresponding to distinct eigenvalues of a symmetric matrix are orthogonal.
3. Diagonalizability: Symmetric matrices are diagonalizable via an orthogonal matrix, i.e., $A=Q\Lambda Q^{\top }$, where $Q$ is Orthogonal matrix and $\Lambda$ is diagonal.
4. Positive Semi-Definite Matrix: If all eigenvalues of $A$ are non-negative, $A$ is positive semidefinite.
5. Closure Under Addition and Scalar Multiplication: The sum of two symmetric matrices and the scalar multiple of a symmetric matrix are symmetric.

Applications

• Quadratic Forms: Symmetric matrices represent quadratic forms, which appear in optimization and geometry.
• Spectral Theorems: Used in principal component analysis (PCA) and other Eigendecomposition-based methods.