Outline of linear algebra

This is a list of linear algebra topics. See also:

• List of matrices
• Glossary of tensor theory.

Linear equations

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

Matrices

• 2 × 2 real matrices
• Matrix theory
• 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

• Cholesky decomposition
• LU decomposition
• QR decomposition
• Polar decomposition
• 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

• 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

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

Affine space and related topics

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

Projective space

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