Linear and multilinear algebra; matrix theory
685 ideas in this subject. The map below lists the main ideas of the subject by area, in teaching order. 244 of its 248 ideas have a page so far.
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- ConceptVector space
A vector space over a field is a set of vectors that can be added and multiplied by scalars, with the usual rules of arithmetic. Vector spaces are the main objects of linear algebra.
- ConceptLinear subspace
A linear subspace of a vector space is a subset that contains the zero vector and is closed under addition and scalar multiplication. It is a vector space with the same operations.
- ConceptLinear combination
A linear combination of vectors is a sum of scalar multiples of the vectors.
- ConceptLinear span
The linear span of a set of vectors is the set of all linear combinations of the vectors. It is the smallest subspace that contains the set.
- ConceptLinear independence
Vectors are linearly independent when no nontrivial linear combination of them is zero. Equivalently, no vector of the family is a linear combination of the others.
- ConceptBasis
A basis of a vector space is a linearly independent family of vectors that spans the space. Each vector of the space is then a linear combination of the basis vectors in exactly one way.
Map of the subject
Vector spaces 19 of 20
- Vector space definition
- Linear subspace definition
- Linear combination definition
- Linear span definition
- Linear independence definition
- Basis definition
- Every vector space has a basis statement
- Every linearly independent set extends to a basis statement
- Steinitz exchange lemma statement
- Dimension definition
- Dimension definition
- Finite-dimensional vector space definition
- Coordinate vector definition
- Change of basis matrix concept
- External direct sum definition
- quotient space definition
- Dimension of a quotient space statement
- Grassmann's formula statement
- Complementary subspaces definition
- Affine subspace definition
Linear maps 17 of 17
- Linear map definition
- Linear isomorphism definition
- Kernel of a linear map definition
- Image of a linear map definition
- Rank–nullity theorem statement
- First isomorphism theorem for linear maps statement
- Matrix of a linear map definition
- Linear map of a matrix definition
- Endomorphism definition
- Projection definition
- Nilpotent endomorphism definition
- Invariant subspace definition
- Dual space definition
- Dual basis definition
- Canonical isomorphism with the double dual statement
- Annihilator of a subspace definition
- Dual map definition
Matrices and matrix algebra 21 of 22
- Matrix definition
- Matrix multiplication definition
- Matrix–vector product definition
- Dot product definition
- Cross product definition
- identity matrix definition
- Transpose definition
- Conjugate transpose definition
- Trace of a matrix definition
- Trace of a product is symmetric statement
- Invertible matrix definition
- A matrix is invertible iff its determinant is a unit statement
- Block matrix definition
- Schur complement definition
- Kronecker product definition
- Hadamard product definition
- Elementary matrix definition
- Reduction to diagonal form by transvections statement
- Similar matrices definition
- Matrix congruence definition
- Equivalent matrices definition
- Commuting matrices concept
Systems of linear equations 11 of 11
- System of linear equations definition
- Gaussian elimination concept
- Row echelon form definition
- Rank of a matrix definition
- Row rank equals column rank statement
- Rank of a product of matrices statement
- Rouché–Capelli theorem statement
- Cramer's rule statement
- Fredholm alternative statement
- Overdetermined system definition
- Least squares concept
Determinants 22 of 22
- Determinant definition
- Leibniz formula for determinants statement
- Laplace expansion along a row statement
- Determinant is multiplicative statement
- Determinant of a transpose statement
- Determinant of a triangular matrix statement
- Determinant via the Schur complement statement
- Cauchy–Binet formula statement
- Adjugate matrix definition
- Minor definition
- Vandermonde determinant statement
- Sylvester's determinant identity statement
- Matrix determinant lemma statement
- Jacobi's formula statement
- Dodgson condensation concept
- Hadamard's maximal determinant problem problem
- Hadamard's inequality statement
- Pfaffian definition
- Permanent definition
- van der Waerden conjecture on permanents statement
- Lindström–Gessel–Viennot lemma statement
- Determinant of a linear map definition
Eigenvalues and eigenvectors 22 of 22
- Eigenvalue definition
- Eigenspace definition
- Generalised eigenvector definition
- Eigenvectors with distinct eigenvalues are linearly independent statement
- Existence of an eigenvalue over an algebraically closed field statement
- Characteristic polynomial definition
- Cayley–Hamilton theorem statement
- Minimal polynomial definition
- Minimal polynomial divides the characteristic polynomial statement
- Algebraic multiplicity definition
- Determinant is the product of the eigenvalues statement
- Trace is the sum of the eigenvalues statement
- Spectrum of a matrix definition
- Spectral radius definition
- Gershgorin circle theorem statement
- Levy–Desplanques theorem statement
- Rayleigh quotient definition
- Courant–Fischer min-max theorem statement
- Cauchy interlacing theorem statement
- Weyl's inequality statement
- Bauer–Fike theorem statement
- Eigenvalue perturbation concept
Canonical forms and decompositions 22 of 23
- Diagonalisable matrix definition
- Triangularisation statement
- Simultaneous diagonalisation statement
- Jordan normal form statement
- Jordan–Chevalley decomposition statement
- Decomposition into generalised eigenspaces statement
- Semisimple endomorphism definition
- Rational canonical form statement
- Companion matrix definition
- Smith normal form statement
- Hermite normal form statement
- Schur decomposition statement
- LU decomposition statement
- QR decomposition statement
- Cholesky decomposition statement
- Singular value decomposition statement
- Singular value definition
- Polar decomposition statement
- Eigendecomposition statement
- Eckart–Young theorem statement
- Moore–Penrose inverse definition
- Generalized inverse definition
- Drazin inverse definition
Inner product spaces 15 of 15
- Inner product space definition
- Cauchy–Schwarz inequality statement
- Orthogonal vectors definition
- Orthonormal basis definition
- Gram–Schmidt process concept
- Orthogonal complement definition
- Orthogonal projection definition
- Bessel's inequality statement
- Gram matrix definition
- Adjoint of a linear map definition
- Self-adjoint operator definition
- Normal matrix definition
- Spectral theorem for Hermitian matrices statement
- Spectral theorem for normal matrices statement
- Linear isometry definition
Bilinear and quadratic forms 15 of 15
- Bilinear form definition
- Nondegenerate bilinear form definition
- Symmetric bilinear form definition
- Alternating bilinear form definition
- Sesquilinear form definition
- Quadratic form definition
- Polarisation identity statement
- Diagonalisation of a quadratic form statement
- Sylvester's law of inertia statement
- Signature of a quadratic form definition
- Sylvester's criterion statement
- Witt's theorem statement
- Witt group definition
- Symplectic vector space definition
- Hermitian form definition
Special matrices 33 of 34
- Diagonal matrix definition
- Triangular matrix definition
- Symmetric matrix definition
- Skew-symmetric matrix definition
- Hermitian matrix definition
- Positive definite matrix definition
- Positive semidefinite matrix definition
- Eigenvalues of a positive definite matrix are positive statement
- Orthogonal matrix definition
- Unitary matrix definition
- Permutation matrix definition
- Idempotent matrix definition
- Nilpotent matrix definition
- Vandermonde matrix definition
- circulant matrix definition
- Toeplitz matrix definition
- Hankel matrix definition
- Hilbert matrix example
- Cauchy matrix definition
- Hadamard matrix definition
- Hadamard conjecture conjecture
- Hessenberg matrix definition
- Tridiagonal matrix definition
- Band matrix definition
- Sparse matrix definition
- Diagonally dominant matrix definition
- Adjacency matrix definition
- Laplacian matrix definition
- Incidence matrix definition
- Totally unimodular matrix definition
- Unimodular matrix definition
- Totally positive matrix definition
- M-matrix definition
- Random matrix concept
Nonnegative and stochastic matrices 8 of 8
- Nonnegative matrix definition
- Perron–Frobenius theorem statement
- Irreducible matrix definition
- Transition matrix definition
- Doubly stochastic matrix definition
- Birkhoff–von Neumann theorem statement
- Majorization definition
- Schur–Horn theorem statement
Matrix analysis 14 of 14
- Matrix norm definition
- Frobenius norm definition
- Operator norm definition
- Condition number definition
- Matrix exponential definition
- Determinant of a matrix exponential statement
- Logarithm of a matrix definition
- Square root of a matrix definition
- Matrix function concept
- Gelfand's formula statement
- Lie product formula statement
- Golden–Thompson inequality statement
- Loewner order definition
- Schur product theorem statement
Matrix equations and groups 10 of 10
- Sylvester equation definition
- Lyapunov equation definition
- Algebraic Riccati equation definition
- General linear group object
- Special linear group object
- Orthogonal group object
- Unitary group object
- Symplectic group object
- Matrix pencil definition
- Matrix completion problem
Multilinear algebra 8 of 8
- Multilinear map definition
- Tensor product of vector spaces definition
- Tensor algebra definition
- Exterior algebra definition
- Symmetric algebra definition
- Clifford algebra definition
- Tensor rank definition
- Tensor decomposition concept
Linear algebra over rings and numerical aspects 7 of 7
- Max-plus algebra concept
- Boolean matrix definition
- Numerical linear algebra concept
- Power iteration concept
- QR algorithm concept
- Strassen algorithm concept
- Matrix multiplication exponent problem
More ideas in this subject
Concepts 94
- Back substitution
- Computing the inverse by row reduction
- Definite and indefinite quadratic forms
- Gauss–Jordan elimination
- Linear equation
- LU decomposition
- Conformal geometric algebra
- Elementary row operations
- Fourier–Motzkin elimination
- Inner product
- Matrix addition
- Product of vector spaces
- Scalar multiplication of matrices
- Square matrix
- Wedge product of forms
- Block diagonal matrix
- Clifford module
- Composition of linear maps
- Consistent and inconsistent systems
- Homogeneous system of linear equations
- Hurwitz-stable matrix
- Jordan block
- Linear form
- Lower triangular matrix
- Orthogonal family for a bilinear form
- Pivot
- Quotient space
- Reduced row echelon form
- Restricted isometry property
- Row equivalence
- Space of linear maps
- Sum of subspaces
- Upper triangular matrix
- Affine independence
- Alternating multilinear map
- Carleman matrix
- Coefficient matrix
- Direct sum of subspaces
- Eigenvalue algorithm
- Euclidean distance matrix
- Orthonormal family
- Polynomial of an endomorphism
- Positive definite quadratic form
- Quotient map
- Reflection
- Spinor bundle
- Tracy–Widom distribution
- Annihilating polynomial
- Augmented matrix
- Berezin integral
- Eigenvector
- Geometric algebra
- Multiplication by a matrix as a linear map
- Null space
- Polar form of a quadratic form
- Scalar matrix
- Spanning set
- Tensor product of algebras
- Transvection
- Affine group
- Column space
- Hodge star operator
- Krylov subspace
- Pseudospectrum
- Restriction of an endomorphism to an invariant subspace
- Row space
- Symmetric bilinear form associated with a quadratic form
- Tensor product of modules
- Normal operator
- Square root of a positive semidefinite matrix
- Stein equation
- Trace of a linear map
- Dirac operator on a spin manifold
- Generalised eigenspace
- Givens rotation
- Determinant as signed volume
- Dirac bundle
- Discrete-time algebraic Riccati equation
- Matrix of a bilinear form
- Powers of a diagonalisable matrix
- Cofactor
- Nullity
- Rank of a linear map
- Clifford analysis
- Dirac operator
- Determinant of a family of vectors in a basis
- Geometric multiplicity
- Orientation of a real vector space
- Characteristic polynomial of a linear map
- Householder transformation
- Tensor product of fields
- Principal component analysis
- Operator monotone function
- Singular values of a compact operator
Theorems and results 102
- Anticommutativity of the cross product
- Scalar triple product
- Vector triple product
- Associativity of matrix multiplication
- Intersection of subspaces
- A linear map is injective iff its kernel is trivial
- Distributivity of matrix multiplication
- Exponential of a sum of commuting matrices
- Farkas' lemma
- Marchenko–Pastur law
- Selberg integral
- Third isomorphism theorem for vector spaces
- Toeplitz–Hausdorff theorem
- Trace is linear
- Transpose of a product
- Cayley–Hamilton theorem for modules
- Choi's theorem on completely positive maps
- Löwner's theorem
- Schoenberg's embedding criterion
- Szegő limit theorems
- Trace of a transpose
- Universal property of the tensor product
- Existence of a complementary subspace
- Inverse of a product
- One-sided inverse of a square matrix is two-sided
- Riesz representation theorem in finite dimensions
- Second isomorphism theorem for vector spaces
- Tensor-hom adjunction
- The cross product is perpendicular to its factors
- Universal property of the quotient space
- An orthonormal family is linearly independent
- Gram matrices are positive semidefinite
- Minimal polynomial divides every annihilating polynomial
- Positivity of Hankel matrices of moments
- Dimension theorem for vector spaces
- Eigenvalues are the roots of the minimal polynomial
- Every spanning set contains a basis
- A linear map is determined by its values on a basis
- Adding a multiple of one row to another preserves the determinant
- Determinant is multilinear in the rows
- Determinant of a 2 by 2 matrix
- Determinant of a matrix with a zero row
- Determinant of a matrix with two equal rows
- Determinant of the identity matrix
- Eigenvalues of a self-adjoint operator are real
- Levinson recursion
- Permuting rows multiplies the determinant by the sign
- Rule of Sarrus
- Scaling a row scales the determinant
- Sylvester's determinant theorem
- Wigner semicircle law
- A linear isometry preserves the inner product
- A subspace of full dimension is the whole space
- Basis of a tensor product
- Determinant is a coefficient of the characteristic polynomial
- Determinant of a block triangular matrix
- Determinant of a scalar multiple
- Dimension of a subspace
- Gram–Schmidt vectors are orthogonal
- Linearly independent set of the right size is a basis
- Dimension of a direct product
- Dimension of the coordinate space
- Isomorphism with the coordinate space
- Laplace expansion along a column
- Matrix exponential as the solution of a linear system
- Trace is a coefficient of the characteristic polynomial
- Change of basis for a bilinear form
- Change of basis formula
- Coordinates in an orthonormal basis
- Determinant of the inverse
- Dimension of the space of linear maps
- Finite-dimensional spaces are isomorphic iff they have equal dimension
- Characteristic polynomial does not depend on the basis
- Dimension of a tensor product
- Dimension of the dual space
- Dimension of the orthogonal complement
- Eigenspaces of a self-adjoint operator are orthogonal
- Existence of an orthonormal basis
- Product of a matrix and its adjugate
- Dimension of the solution space of a linear recurrence
- Matrix of a composition is the product of matrices
- Diagonalisable iff the minimal polynomial has distinct roots
- Dimension of the annihilator
- An injective endomorphism of a finite-dimensional space is surjective
- Inverse in terms of the adjugate
- Trace of a linear map is the trace of its matrix
- Classification of complex quadratic forms
- Bareiss algorithm
- Cartan–Dieudonné theorem
- Matrix of the dual map is the transpose
- Determinant of a linear map is the determinant of its matrix
- Hadamard's inequality in Fredholm theory
- Spectral theorem
- A matrix is diagonalisable iff it has a basis of eigenvectors
- Amitsur–Levitzki theorem
- Matrix of the adjoint is the conjugate transpose
- Cayley–Hamilton theorem for linear maps
- Eigenvalues are the roots of the characteristic polynomial
- Distinct eigenvalues imply diagonalisable
- Principal axis theorem
- Mehta integral
- Schmidt's singular value decomposition of a kernel
Examples 14
- Function space
- Space of polynomials of bounded degree
- Circulant matrix
- Determinant as a group homomorphism
- Differentiation as a linear map
- Space of matrices
- Tropical semiring
- Affine space as an algebra
- Indefinite orthogonal group
- Birkhoff polytope
- Rotation matrix
- Discontinuous additive functions via a Hamel basis
- Spin group
- Pin group
Counterexamples 1
Objects 7
Conjectures 1
Definitions 234
- Linear equation
- Matrix
- Vector space
- Bilinear form
- Block matrix
- Conformal geometric algebra
- Cross product
- Diagonally dominant matrix
- Dot product
- Elementary row operations
- External direct sum
- Fourier–Motzkin elimination
- Inner product
- Linear combination
- Linear map
- Linear subspace
- Majorization
- Matrix addition
- Matrix multiplication
- Matrix–vector product
- Max-plus algebra
- Nonnegative matrix
- Product of vector spaces
- Row echelon form
- Scalar multiplication of matrices
- Sesquilinear form
- Sparse matrix
- Square matrix
- System of linear equations
- Transpose
- Alternating bilinear form
- Band matrix
- Block diagonal matrix
- Cauchy matrix
- Clifford module
- Composition of linear maps
- Conjugate transpose
- Consistent and inconsistent systems
- Diagonal matrix
- Doubly stochastic matrix
- Endomorphism
- Generalized inverse
- Gram matrix
- Hankel matrix
- Hessenberg matrix
- Homogeneous system of linear equations
- Hurwitz-stable matrix
- Idempotent matrix
- Image of a linear map
- Inner product space
- Jordan block
- Kernel of a linear map
- Kronecker product
- Linear form
- Linear independence
- Linear isomorphism
- Logarithm of a matrix
- Lower triangular matrix
- Matrix exponential
- Multilinear map
- Nilpotent matrix
- Nondegenerate bilinear form
- Orthogonal family for a bilinear form
- Orthogonal vectors
- Overdetermined system
- Permutation matrix
- Pivot
- Quadratic form
- Quotient space
- Reduced row echelon form
- Restricted isometry property
- Row equivalence
- Space of linear maps
- Square root of a matrix
- Sum of subspaces
- Symmetric bilinear form
- Tensor algebra
- Tensor product of vector spaces
- Toeplitz matrix
- Totally positive matrix
- Totally unimodular matrix
- Trace of a matrix
- Tridiagonal matrix
- Upper triangular matrix
- Vandermonde matrix
- Affine independence
- Alternating multilinear map
- Boolean matrix
- Carleman matrix
- Coefficient matrix
- Direct sum of subspaces
- Dodgson condensation
- Drazin inverse
- Eigenvalue
- Eigenvalue algorithm
- Elementary matrix
- Euclidean distance matrix
- Exterior algebra
- Hermitian form
- Irreducible matrix
- Linear span
- Matrix function
- Matrix norm
- Moore–Penrose inverse
- Nilpotent endomorphism
- Normal matrix
- Orthogonal matrix
- Orthonormal family
- Permanent
- Polynomial of an endomorphism
- Positive definite quadratic form
- Quotient map
- Random matrix
- Skew-symmetric matrix
- Spectral radius
- Spinor bundle
- Symmetric matrix
- Tensor decomposition
- Tensor rank
- Tracy–Widom distribution
- Unimodular matrix
- Adjoint of a linear map
- Annihilating polynomial
- Augmented matrix
- Berezin integral
- Complementary subspaces
- Condition number
- Dual space
- Eigenvector
- Geometric algebra
- Hadamard product
- Hermitian matrix
- Invariant subspace
- Invertible matrix
- Linear isometry
- Loewner order
- Lyapunov equation
- M-matrix
- Matrix pencil
- Multiplication by a matrix as a linear map
- Null space
- Orthogonal complement
- Polar form of a quadratic form
- Power iteration
- Scalar matrix
- Spanning set
- Spectrum of a matrix
- Strassen algorithm
- Sylvester equation
- Symmetric algebra
- Symplectic vector space
- Tensor product of algebras
- Transvection
- Triangular matrix
- Unitary matrix
- Affine group
- Algebraic Riccati equation
- Clifford algebra
- Column space
- Dual map
- Equivalent matrices
- Finite-dimensional vector space
- Frobenius norm
- Hodge star operator
- Krylov subspace
- Minimal polynomial
- Pfaffian
- Positive semidefinite matrix
- Pseudospectrum
- Restriction of an endomorphism to an invariant subspace
- Row space
- Semisimple endomorphism
- Similar matrices
- Symmetric bilinear form associated with a quadratic form
- Tensor product of modules
- Annihilator of a subspace
- Basis
- Commuting matrices
- Eigenspace
- Generalised eigenvector
- Normal operator
- Positive definite matrix
- Projection
- Schur complement
- Self-adjoint operator
- Square root of a positive semidefinite matrix
- Stein equation
- Trace of a linear map
- Coordinate vector
- Determinant
- Diagonalisable matrix
- Dirac operator on a spin manifold
- Generalised eigenspace
- Givens rotation
- Change of basis matrix
- Characteristic polynomial
- Dimension
- Dirac bundle
- Discrete-time algebraic Riccati equation
- Matrix of a bilinear form
- Minor
- Witt group
- Cofactor
- Companion matrix
- Eigenvalue perturbation
- Matrix of a linear map
- Rayleigh quotient
- Linear map of a matrix
- Orthonormal basis
- Adjugate matrix
- Dual basis
- Matrix congruence
- Nullity
- Orthogonal projection
- Rank of a linear map
- Rank of a matrix
- Singular value
- Reflection
- Wedge product of forms
- Clifford analysis
- Dirac operator
- Determinant of a family of vectors in a basis
- Geometric multiplicity
- Orientation of a real vector space
- Determinant of a linear map
- Characteristic polynomial of a linear map
- Signature of a quadratic form
- Householder transformation
- Tensor product of fields
- Algebraic multiplicity
- QR algorithm
- Principal component analysis
- Operator monotone function
- Singular values of a compact operator