# ⓘ List of linear algebra topics

## ⓘ List of linear algebra topics

• Basis transformation matrix
• Weyr canonical form
• Matrix theory
• Trace
• Jordan normal form
• Characteristic polynomial
• Matrix multiplication
• Eigenvalue, eigenvector and eigenspace
• 2 × 2 real matrices
• Cayley–Hamilton theorem
• Pseudoinverse
• Matrix inversion, invertible matrix
• Rank
• Dot product
• Hermitian matrix, Antihermitian matrix
• Symmetric matrix
• Unitary matrix
• Conjugate transpose
• Skew-symmetric matrix
• Orthogonal matrix
• Transpose
• Spectral theorem
• List of matrices
• Projection
• Perron–Frobenius theorem
• Positive-definite, positive-semidefinite matrix
• Diagonalizable matrix
• Pfaffian
• Diagonal matrix, main diagonal
• Hessenberg matrix
• Tridiagonal matrix
• Hessian matrix
• Block matrix
• Circulant matrix
• Vandermonde matrix
• Sparse matrix
• Toeplitz matrix
• Triangular matrix
• Stochastic matrix
• 0.1-matrix
• Hankel matrix

## 1. Matrix decompositions

• Spectral theorem
• QR decomposition
• Polar decomposition
• Cholesky decomposition
• Higher-order singular value decomposition
• LU decomposition
• Singular value decomposition
• Schur complement
• Schur decomposition

## 2. Vector spaces

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

## 3. Structures

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

## 4. Multilinear algebra

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

## 5. Affine space and related topics

• Poincare group
• Affine group
• Flat geometry
• Affine transformation
• Affine coordinate system
• Cartesian coordinate system
• Affine geometry
• Galilean group
• Euclidean group

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