Eigenvalues and Dynamics
Cheat sheet.

The key rules, formulas and reminders from all 4 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Eigenvalues and Eigenspaces

Av=\lambda v,\ v\ne0;

\det(A-\lambda I)=0;

E_\lambda=N(A-\lambda I).

An eigenvalue may be zero, but an eigenvector may not. Find the null space for each eigenvalue.

A=VDV^{-1}\Rightarrow A^k=VD^kV^{-1}.

Diagonalization requires a full independent eigenvector basis. Repeated eigenvalues neither guarantee nor rule it out.

i^2=-1,\quad |a+ib|=\sqrt{a^2+b^2},\quad (re^{i\theta})^k=r^ke^{ik\theta}.

Real matrices may require complex eigenvectors. For complex inner products, use conjugate transpose.

Real symmetric $A$ has $A=Q\Lambda Q^T$ . In eigen-coordinates, $x^TAx=\sum\lambda_i y_i^2$ .

Positive semidefinite allows zero directions. A stationary point is a strict quadratic minimum only with positive definite curvature.

2 reference blocks

Av=\lambda v,\ v\ne0;

\det(A-\lambda I)=0;

E_\lambda=N(A-\lambda I).

An eigenvalue may be zero, but an eigenvector may not. Find the null space for each eigenvalue.

2 reference blocks

A=VDV^{-1}\Rightarrow A^k=VD^kV^{-1}.

Diagonalization requires a full independent eigenvector basis. Repeated eigenvalues neither guarantee nor rule it out.

2 reference blocks

i^2=-1,\quad |a+ib|=\sqrt{a^2+b^2},\quad (re^{i\theta})^k=r^ke^{ik\theta}.

Real matrices may require complex eigenvectors. For complex inner products, use conjugate transpose.

2 reference blocks

Real symmetric $A$ has $A=Q\Lambda Q^T$ . In eigen-coordinates, $x^TAx=\sum\lambda_i y_i^2$ .

Positive semidefinite allows zero directions. A stationary point is a strict quadratic minimum only with positive definite curvature.