Key Concepts

 Data Analytics: Summaries on Strang's Linear Algebra - Symmetric Matrices
Written by FemiByte
Tuesday, 05 March 2013 06:09

Symmetric Matrices

Properties

1. A symmetric matrix has only real eigenvalues.
2. The eigenvectors can be chosen orthonormal.

Spectral Theorem

Every symmetric matrix has the factorization $A=Q\Lambda Q^T$ with real eigenvalues in $\Lambda$ and orthonormal eigenvectors in $S=Q$:
Symmetric diagonalization: $A = Q\Lambda Q^{-1} = Q \Lambda Q^T \text{ with } Q^{-1}=Q^T$ Real eigenvalues - All the eigenvalues of a real symmetric matrix are real.
Orthogonal Eigenvectors - Eigenvectors of a real symmetric matrix (when then correspond to different $\lambda$'s) are always perpendicular.

Eigenvalues vs. Pivots

If $A$ is symmetric, the number of +ve eigenvalues of $A=A^T$ equals the number of +ve pivots. Corresponding pivots and eigenvalues have matching signs when $A$ is symmetric. All symmetric matrices are diagonalizable. There are always enough eigenvectors to diagonalize $A=A^T$.

Schur's Theorem:Every square matrix factors into $A=QTQ^{-1}$ where $T$ is upper triangular and $\bar{Q}^T=Q^{-1}$. If $A$ has real eigenvalues then $Q$ and $T$ can be chosen real: $Q^TQ=I$.

Key Facts

1. A symmetric matrix has real eigenvalues and perpendicular eigenvectors.
2. Diagonalization becomes $A=Q\Lambda Q^T$ with an orthogonal matrix $Q$.
3. All symmetric matrices are diagonalizable, even with repeated eigenvalues.
4. The signs of the eigenvalues match the songs of the pivots, when $A=A^T$.
5. Every square matrix can be "triangularized" by $A=QTQ^{-1}$.

Last Updated on Tuesday, 05 March 2013 06:17