15Axx Linear and multilinear algebra
This subtopic studies linear and multilinear algebra, covering vector spaces, tensors, linear maps, and the structural principles that govern these systems.
Specific topics
15A03 Vector spaces, linear dependence, rank, lineability
Overview
15A03 studies vector spaces, linear dependence, rank, lineability in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Vector spaces, linear dependence, rank, lineability (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for vector spaces, linear dependence, rank, lineability
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A04 Linear transformations, semilinear transformations
Overview
15A04 studies linear transformations, semilinear transformations in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Linear transformations, semilinear transformations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for linear transformations, semilinear transformations
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A06 Linear equations
Overview
15A06 studies linear equations in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Linear equations (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for linear equations
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A09 Theory of matrix inversion and generalized inverses
Overview
15A09 studies theory of matrix inversion and generalized inverses in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Theory of matrix inversion and generalized inverses (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for theory of matrix inversion and generalized inverses
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A12 Conditioning of matrices
Overview
15A12 studies conditioning of matrices in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Conditioning of matrices (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for conditioning of matrices
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A15 Determinants, permanents, traces, other special matrix functions
Overview
15A15 studies determinants, permanents, traces, other special matrix functions in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Determinants, permanents, traces, other special matrix functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for determinants, permanents, traces, other special matrix functions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A16 Matrix exponential and related functions of matrices
Overview
15A16 studies matrix exponential and related functions of matrices in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Matrix exponential and related functions of matrices (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for matrix exponential and related functions of matrices
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A18 Eigenvalues, singular values, and eigenvectors
Overview
15A18 studies eigenvalues, singular values, and eigenvectors in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Eigenvalues, singular values, and eigenvectors (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for eigenvalues, singular values, and eigenvectors
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A21 Canonical forms, reductions, classification
Overview
15A21 studies canonical forms, reductions, classification in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Canonical forms, reductions, classification (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for canonical forms, reductions, classification
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A22 Matrix pencils
Overview
15A22 studies matrix pencils in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Matrix pencils (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for matrix pencils
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A23 Factorization of matrices
Overview
15A23 studies factorization of matrices in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Factorization of matrices (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for factorization of matrices
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A24 Matrix equations and identities
Overview
15A24 studies matrix equations and identities in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Matrix equations and identities (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for matrix equations and identities
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A27 Commutativity of matrices
Overview
15A27 studies commutativity of matrices in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Commutativity of matrices (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for commutativity of matrices
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A29 Inverse problems in linear algebra
Overview
15A29 studies inverse problems in linear algebra in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Inverse problems in linear algebra (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for inverse problems in linear algebra
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A39 Linear inequalities of matrices
Overview
15A39 studies linear inequalities of matrices in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Linear inequalities of matrices (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for linear inequalities of matrices
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A42 Inequalities involving eigenvalues and eigenvectors
Overview
15A42 studies inequalities involving eigenvalues and eigenvectors in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Inequalities involving eigenvalues and eigenvectors (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for inequalities involving eigenvalues and eigenvectors
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A45 Miscellaneous inequalities involving matrices
Overview
15A45 studies miscellaneous inequalities involving matrices in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Miscellaneous inequalities involving matrices (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for miscellaneous inequalities involving matrices
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A54 Matrices over function rings
Overview
15A54 studies matrices over function rings in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Matrices over function rings (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for matrices over function rings
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A60 Norms of matrices, numerical range, applications of functional analysis
Overview
15A60 studies norms of matrices, numerical range, applications of functional analysis in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Norms of matrices, numerical range, applications of functional analysis (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for norms of matrices, numerical range, applications of functional analysis
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A63 Quadratic and bilinear forms, inner products
Overview
15A63 studies quadratic and bilinear forms, inner products in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Quadratic and bilinear forms, inner products (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for quadratic and bilinear forms, inner products
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A66 Clifford algebras, spinors
Overview
15A66 studies clifford algebras, spinors in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Clifford algebras, spinors (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for clifford algebras, spinors
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A69 Multilinear algebra, tensor calculus
Overview
15A69 studies multilinear algebra, tensor calculus in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Multilinear algebra, tensor calculus (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for multilinear algebra, tensor calculus
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A72 Vector and tensor algebra, theory of invariants
Overview
15A72 studies vector and tensor algebra, theory of invariants in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Vector and tensor algebra, theory of invariants (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for vector and tensor algebra, theory of invariants
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A75 Exterior algebra, Grassmann algebras
Overview
15A75 studies exterior algebra, grassmann algebras in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Exterior algebra, Grassmann algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for exterior algebra, grassmann algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A78 Other algebras built from modules
Overview
15A78 studies other algebras built from modules in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Other algebras built from modules (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for other algebras built from modules
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A80 Max-plus and related algebras
Overview
15A80 studies max-plus and related algebras in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Max-plus and related algebras (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for max-plus and related algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A83 Matrix completion problems
Overview
15A83 studies matrix completion problems in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Matrix completion problems (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for matrix completion problems
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks
15A86 Linear preserver problems
Overview
15A86 studies linear preserver problems in linear and multilinear algebra. It covers the structural and computational theory of linear maps, matrices, tensors, and multilinear constructions used throughout mathematics and its applications.
Related Wikipedia Page
Linear preserver problems (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for linear preserver problems
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used to organize definitions, compare equivalent formulations, and support proof-driven or computational work in algebra and algebraic geometry.
Applications
- Linear algebra foundations and matrix analysis
- Tensor, invariant, and multilinear methods
- Applications in optimization, data science, and operator theory
References
Recommended Textbooks