Outline of linear algebra explained

This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices.

Linear equations

Linear equation

• System of linear equations
• Determinant
  • Minor
  • Cauchy–Binet formula
• Cramer's rule
• Gaussian elimination
• Gauss–Jordan elimination
• Overcompleteness
• Strassen algorithm

Matrices

Matrix

• Matrix addition
• Matrix multiplication
• Basis transformation matrix
• Characteristic polynomial
• Trace
• Eigenvalue, eigenvector and eigenspace
  • Cayley–Hamilton theorem
  • Spread of a matrix
  • Jordan normal form
  • Weyr canonical form
• Rank
• Matrix inversion, invertible matrix
  • Pseudoinverse
• Adjugate
• Transpose
  • Dot product
  • Symmetric matrix
  • Orthogonal matrix
  • Skew-symmetric matrix
  • Conjugate transpose
    • Unitary matrix
    • Hermitian matrix, Antihermitian matrix
• Positive-definite, positive-semidefinite matrix
• Pfaffian
• Projection
• Spectral theorem
• Perron–Frobenius theorem
• List of matrices
  • Diagonal matrix, main diagonal
    • Diagonalizable matrix
  • Triangular matrix
  • Tridiagonal matrix
  • Block matrix
  • Sparse matrix
  • Hessenberg matrix
  • Hessian matrix
  • Vandermonde matrix
  • Stochastic matrix
  • Toeplitz matrix
    • Circulant matrix
  • Hankel matrix
  • (0,1)-matrix

Matrix decompositions

Matrix decomposition

• Cholesky decomposition
• LU decomposition
• QR decomposition
• Polar decomposition
• Reducing subspace
• Spectral theorem
• Singular value decomposition
  • Higher-order singular value decomposition
• Schur decomposition
  • Schur complement
  • Haynsworth inertia additivity formula

Relations

• Matrix equivalence
• Matrix congruence
• Matrix similarity
• Matrix consimilarity
• Row equivalence

Computations

• Elementary row operations
• Householder transformation
• Least squares, linear least squares
• Gram–Schmidt process
• Woodbury matrix identity

Vector spaces

Vector space

• Linear combination
• Linear span
• Linear independence
• Scalar multiplication
• Basis
  • Change of basis
  • Hamel basis
• Cyclic decomposition theorem
• Dimension theorem for vector spaces
  • Hamel dimension
• Examples of vector spaces
• Linear map
  • Shear mapping or Galilean transformation
  • Squeeze mapping or Lorentz transformation
• Linear subspace
  • Row and column spaces
  • Column space
  • Row space
  • Cyclic subspace
  • Null space, nullity
  • Rank–nullity theorem
  • Nullity theorem
• Dual space
  • Linear function
  • Linear functional
• Category of vector spaces

Structures

• Topological vector space
• Normed vector space
• Inner product space
  • Euclidean space
  • Orthogonality
  • Orthogonal complement
  • Orthogonal projection
  • Orthogonal group
• Pseudo-Euclidean space
  • Null vector
  • Indefinite orthogonal group
• Orientation (geometry)
  • Improper rotation
• Symplectic structure

Multilinear algebra

Multilinear algebra

• Tensor
  • Classical treatment of tensors
  • Component-free treatment of tensors
  • Gamas's Theorem
• Outer product
• Tensor algebra
  • Exterior algebra
  • Symmetric algebra
  • Clifford algebra
  • Geometric algebra

Topics related to affine spaces

Affine space

• Affine transformation
• Affine group
• Affine geometry
• Affine coordinate system
• Flat (geometry)
• Cartesian coordinate system
• Euclidean group
• Poincaré group
• Galilean group

Projective space

Projective space

• Projective transformation
• Projective geometry
• Projective linear group
• Quadric and conic section

See also

• Glossary of linear algebra
• Glossary of tensor theory