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Eigenvalues and Eigenvectors - Deep Dive

Mathematical Foundations​

Rigorous treatment of spectral theory with proofs and applications.

Eigenvalue Theory​

Fundamental Theorems​

Theorem (Spectral Theorem): A real symmetric matrix has real eigenvalues and orthogonal eigenvectors.

Proof: (To be completed)

Characteristic Polynomial​

Theorem: The eigenvalues of A are roots of det(A - λI) = 0.

Proof: (To be completed)

Diagonalization Theory​

Conditions for Diagonalizability​

Theorem: An n×n matrix A is diagonalizable if and only if...

Proof: (To be completed)

Spectral Decomposition​

(To be completed)

Applications​

Matrix Powers​

Theorem: If A = PDP⁻¹, then Aᵏ = PDᵏP⁻¹

Application to Difference Equations: (To be completed)

Matrix Exponentials​

Definition: eᴬᵗ = I + At + (At)²/2! + ...

Theorem: If A = PDP⁻¹, then eᴬᵗ = Peᴰᵗ P⁻¹

Application to Differential Equations: (To be completed)

Complex Eigenvalues​

Theory​

(To be completed)

Exercises​

(Advanced problems to be completed)