Eigendecomposition

An eigenvector $\mathbf{v}$ of matrix $A$ is a nonzero vector whose direction is unchanged by the transformation: $A\mathbf{v} = \lambda\mathbf{v}$, where $\lambda$ is the eigenvalue.

Geometric meaning: eigenvectors are the directions that get purely scaled (not rotated) under the transformation. The eigenvalue $\lambda$ is the scale factor.

Eigendecomposition: if $A$ has $n$ linearly independent eigenvectors, then $A = PDP^{-1}$, where $P$ is the matrix of eigenvectors and $D$ is diagonal with eigenvalues.

Key properties:

• $\text{det}(A) = \prod \lambda_i$ — determinant is the product of eigenvalues
• $\text{tr}(A) = \sum \lambda_i$ — trace is the sum of eigenvalues
• $A^k = PD^kP^{-1}$ — powers become trivial in eigenspace

Where it appears:

• Principal Component Analysis (PCA) — eigenvectors of the covariance matrix are principal directions
• Positive Definite Matrices — characterized by all $\lambda_i > 0$
• Stability analysis — system $\dot{x} = Ax$ is stable iff all eigenvalues have negative real part

Computed via Power Iteration (dominant eigenvalue) or characteristic polynomial $\det(A - \lambda I) = 0$.

See also: Singular Value Decomposition (SVD)