Eigenvalues and Eigenvectors

Eigen-theory Fundamentals

Definition: $A\vec{v}=\lambda \vec{v}$

Characteristic Equation: $\det (A-\lambda I)=0$

Finding Eigenvalues (Spectrum)

Finding Eigenvectors (Eigenspaces $E_{\lambda }$)

Multiplicity

Algebraic Multiplicity (root of characteristic polynomial)

Geometric Multiplicity (dimension of eigenspace)

Diagonalization

Conditions for Diagonalization (Alg. Mult. == Geom. Mult.)

Diagonal Matrix Decomposition: $A=PD{P}^{-1}$

Symmetric Matrices and Orthogonal Diagonalization

Applications

Computing Matrix Powers $A^k$

Discrete Dynamical Systems (Markov Chains)

Fibonacci Sequence matrix form

Quadratic Forms and Conic Sections

Handwritten (some repetitions in the first part)