Symmetric Matrices

A symmetric matrix’ Transpose obeys $A^T=A$

• For any $A$, $A{A}^T$ yields a symmetric matrix

Elimination

Elimination with symmetric matrices is more efficient. When a symmetric matrix is factored to A = LDU, $U=L^T$ such that $A=LD{L}^T$.

Eigenvalues and Eigenvectors

Symmetric matrix’ $\lambda$ and $x$ are always real and orthogonal to each other. Therefore their eigenvector matrices are orthogonal, $Q$.

$$S=X\Lambda X^{-1}=Q\Lambda Q^{-1}\text{with}{Q}^{T}Q=I$$

Pivot and Eigenvalue Relationship

The number of positive eigenvalues in $S=S^T$ equals the number of positive pivots.