Linear Algebra

7.

Eigenvalues and Eigenvectors

7.1.

7.1.1.

Definition of eigenvalues and eigenvectors

7.1.2.

Eigenvalue equation

7.1.3.

Geometric interpretation

7.1.4.

7.1.5.

Linear independence of eigenvectors

7.2.

Computing Eigenvalues and Eigenvectors

7.2.1.

Characteristic polynomial

7.2.2.

Characteristic equation

7.2.3.

Algebraic multiplicity

7.2.4.

Geometric multiplicity

7.2.5.

Eigenvalues of special matrices

7.2.6.

Finding eigenvectors

7.3.

Diagonalization

7.3.1.

Diagonalizable matrices

7.3.2.

Conditions for diagonalization

7.3.3.

Diagonalization process

7.3.4.

Similar matrices

7.3.5.

Powers of diagonalizable matrices

7.3.6.

Non-diagonalizable matrices

7.4.

Applications of Eigenvalue Theory

7.4.1.

Systems of differential equations

7.4.2.

Matrix powers and sequences

7.4.3.

Markov chains and steady states

7.4.4.

Discrete dynamical systems

7.4.5.

Principal component analysis

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