Linear algebra

Vector spaces

Decompositions

LU decomposition

QR decomposition

Singular value decomposition (SVD)

Linear operators

Singular value decomposition

Inner product space

Normed space

Hilbert space

Spectral analysis

Generalized eigenvectors

Tensors

Vector space

Vector in a vector space

Additive operator

Homogeneous operator

Linear operator

Types of operators

. And that’s it. A general matrix encode a mix of scaling, rotation and inversion.

Symmetric matrix

Symmetric operator

\[{{L}^{\top}{} = L}\]

Result

Symmetric operators correspond to a symmetric matrix.

Orthogonal operator

Hermitian operator

Hermitian matrix

Adjoint of a matrix

Adjoint of an operator

Hilbert space

A Hilbert space is a complete inner product space.

Eigenvalue of a linear operator

Eigenvector of a linear operator

Characteristic equation for a linear operator

Multiplicity of an eigenvalue

Spectrum of a matrix

Spectrum of a linear operator

Generalized eigenvector

Multilinear operator

Satisfies homogeneity for all arguments: \[{\alphaf({x}_{1}, {x}_{2}, \dots, {x}_{n}) = f(\alpha{x}_{1}, {x}_{2}, \dots, {x}_{n}) = f({x}_{1}, \alpha{x}_{2}, \dots, {x}_{n}) = \dots{} = f({x}_{1}, {x}_{2}, \dots, \alpha{x}_{n})}\] And satisfies additivity for all arguments: \[{f({x}_{1}, y + z, \dots, {x}_{n}) = f({x}_{1}, y, \dots, {x}_{n}) + f({x}_{1}, z, \dots, {x}_{n})}\]

Kronecker product