Eigenvalues and Eigenvectors¶

Formula¶

\[ Av=\lambda v,\quad v\ne 0 \]

Parameters¶

\(A\): square matrix
\(v\): eigenvector
\(\lambda\): eigenvalue

What it means¶

An eigenvector keeps its direction under \(A\), scaled by eigenvalue \(\lambda\).

What it's used for¶

Stability analysis and dynamical systems.
PCA/spectral methods and matrix factorizations.

Key properties¶

Eigenvalues solve \(\det(A-\lambda I)=0\).
Not every matrix is diagonalizable.

Common gotchas¶

Eigenvectors are defined up to nonzero scaling.
Complex eigenvalues/eigenvectors can arise from real matrices.

Example¶

For \(A=\operatorname{diag}(2,3)\), basis vectors are eigenvectors with eigenvalues 2 and 3.

How to Compute (Pseudocode)¶

Input: square matrix A
Output: eigenvalues lambda and eigenvectors v (when requested)

run an eigensolver (for example, QR-based methods for dense matrices)
extract eigenvalues from the solver output
compute/recover eigenvectors if needed
return eigenvalues (and eigenvectors)

Complexity¶

Time: Typically \(O(n^3)\) for dense eigensolver methods on an \(n\times n\) matrix
Space: \(O(n^2)\) for dense matrix/factor storage
Assumptions: Dense direct eigensolver workflow shown; sparse/iterative methods can compute only a few eigenpairs more cheaply