Eigenvalues and Eigenvectors

The eigenvectors of $A$ do not change direction when you multiply them by $A$. The output $Ax$ is on the same line as the input vector $x$.

$$Ax = \lambda x$$

Multiply again by $A$, to see that $x$ is also an eigenvector of $A^2$:

$$A^k x = \lambda^k x$$

• Sum of eigenvalues will be the trace of $A$
• Product of eigenvalues will be the determinant of $A$
• Symmetric matrices will have real eigenvalues

The eigenvalues of any triangular matrix is its main diagonal elements.

Symmetric positive definite matrices

• All $n$ eigenvalues $\lambda$ of a symmetric matrix $S$ are real numbers
• The $n$ eigenvectors $q$ can be chosen orthogonal (perpendicular to each other)

Spectral theorem: Every real symmetric matrix has the form $S = Q \Lambda Q^\text{T}.$

Positive definite matrices

A positive definite matrix has all positive eigenvalues.

If $\lambda \geq 0$ but not $\lambda \gt 0$ then it is positive semidefinite.

Elimation factors every positive definite $S$ into $A^\text{T}A$ (A is upper triangular)

This is Cholesky factorization $S = A^{\text{T}}A$ with $\sqrt{\text{pivots}}$ on the main diagonal of $A$.