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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 15: Linear and multilinear algebra; matrix theory (finite and infinite)

Introduction

Linear algebra, sometimes disguised as matrix theory, considers sets and functions which preserve linear structure. In practice this includes a very wide portion of mathematics! Thus linear algebra includes axiomatic treatments, computational matters, algebraic structures, and even parts of geometry; moreover, it provides tools used for analyzing differential equations, statistical processes, and even physical phenomena.

History

Here is a paper on Hermann Grassmann and the Creation of Linear Algebra. Further reading:

• T.L. Hankins: Sir William Rowan Hamilton, Johns Hopkins UP, 1980.
• M.J. Crowe: A History of Vector Analysis, U Notre Dame Press, 1967, reprinted by Dover, 1985.

Applications and related fields

In the accompanying diagram the reader might observe a few clusters of related fields, showing both the many parts of linear algebra and the related fields in which many of these themes are extended and applied.

This image slightly hand-edited for clarity.

Classic topics in linear algebra and matrix theory are at the center of the diagram: 15A03: Vector spaces, 15A04: Linear transformations, 15A15: Determinants, and 15A21: Canonical forms (e.g. the Jordan canonical form).

Also crowded near the center of the diagram are several fields concerned with linear spaces and linear transformations, and in some cases the reflection of those ideas in the corresponding matrices. Of particular interest are spaces of functions, the modern setting for differential equations and global analysis. Most of the study of linear algebra in these infinite-dimensional (i.e. topological) spaces is classified separately in the fields of functional analysis including 46: Function Analysis proper, 43: Abstract harmonic analysis, and 47: Operator theory. Here we are concerned with similar perspectives with interesting consequences even for finite-dimensional spaces. In particular one might include 15A60: Matrix norms, 15A57: Hermitian and other classes of matrices, 15A24: Matrix equations and identities, 15A54: Matrices over function rings in one or more variables, 15A42: Inequalities involving eigenvalues, and 15A22: Matrix pencils.

At the far right are several large areas of activity in numerical linear algebra and related topics, typically, the study of individual matrices or transformations between (large-dimensional) real vector spaces. Numerical linear algebra per se (e.g. the determination of fast methods of solving thousands of simultaneous linear equations, the stability of eigenvalue calculations, applications to finite-element methods, sparse matrix techniques) are parts of 65: Numerical Analysis, particularly 65F: Numerical linear algebra. Here we include several fields of inquiry into the underlying linear systems and their applications. In this category we might include 15A06: Linear equations, 15A09: Matrix inversion and generalized inverses, 15A18: Eigenvalues and singular values, 15A23: Factorization of matrices (SVD, LU, QR, etc.), 15A12: Conditioning of matrices, as well as applications to physics (15A90), Control Theory (93) and Statistics (62) such as what is there known as Principal Component Analysis. Related topics include those of importance in 90: Operations Research (especially linear programming) such as 15A48: Positive matrices, 15A39: Linear inequalities, and 15A45: Miscellaneous matrix inequalities.

The circles in shades of red in the lower part of the graph show connections with other fields of algebra. Furthest down is 15A72: Invariant theory and tensor algebra, which crosses to the study of invariants in Group Theory (20) and in polynomial rings (13: Commutative Algebra and 14: Algebraic Geometry). Certain sets of matrices form well-known groups, particularly the Lie groups (22) and algebraic groups. Closer to the center of the picture are connections with Number Theory (10 and 11), especially 15A63: Quadratic forms. There are several connections with ring theory (16: Noncommutative Rings, 17: Nonassociative Rings, 19: Algebraic K-Theory); indeed many of the key examples of such rings involve collections of matrices, including the full matrix rings and Lie rings, and rings of matrices are used for representing groups and general rings. Related disciplines within Linear Algebra include 15A27: Commutativity, 15A30: Algebraic systems of matrices, 15A33: Matrices over special rings (including 12: Fields), 15A36: Matrices of integers, 15A75: Grassmann algebras, and 15A78: Other algebras. Tensor products in linear algebra (15A69) mirror such constructs in other algebraic categories.

Nearby are several fields in discrete mathematics, including the use of matrices for the representation of combinatorical objects such as graphs (05: Combinatorics), extremal matrices, permanents (15A15), and applications to 68:Computer Science, 94: Information Theory (e.g. linear codes), and 39: Difference equations.

In the upper left are the papers in "geometric algebra", including 15A66 (Clifford algebras), 81 (Quantum theory), 53 (Differential Geometry), and 58 (Analysis on manifolds).

In the upper right are the topics appropriate for 60: Probability and 62: Statistics, including 15A51: Stochastic matrices and 15A52: Random matrices, and applications to statistical mechanics (82) and the sciences (92).

Linear maps of geometric interest are considered in the geometry pages (51, 52). For example, rotation matrices and affine changes of coordinates come under 51F15. Sets of matrices qua sets arise geometrically as well; for example certain families of matrices form manifolds, and even topological groups (22).

Subfields

There is only one division (15A) but it is subdivided:

• 15A03: Vector spaces, linear dependence, rank
• 15A04: Linear transformations, semilinear transformations
• 15A06: Linear equations
• 15A09: Matrix inversion, generalized inverses
• 15A12: Conditioning of matrices, See also 65F35
• 15A15: Determinants, permanents, other special matrix functions, See also 19B10, 19B14
• 15A18: Eigenvalues, singular values, and eigenvectors
• 15A21: Canonical forms, reductions, classification
• 15A22: Matrix pencils, See also 47A56
• 15A23: Factorization of matrices
• 15A24: Matrix equations and identities
• 15A27: Commutativity
• 15A29: Inverse problems [new in 2000]
• 15A30: Algebraic systems of matrices, See also 16S50, 20Gxx, 20Hxx
• 15A33: Matrices over special rings (quaternions, finite fields, etc.)
• 15A36: Matrices of integers, See also 11C20
• 15A39: Linear inequalities
• 15A42: Inequalities involving eigenvalues and eigenvectors
• 15A45: Miscellaneous inequalities involving matrices
• 15A48: Positive matrices and their generalizations; cones of matrices
• 15A51: Stochastic matrices
• 15A52: Random matrices
• 15A54: Matrices over function rings in one or more variables
• 15A57: Other types of matrices (Hermitian, skew-Hermitian, etc.)
• 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory, See also 65F35, 65J05
• 15A63: Quadratic and bilinear forms, inner products See mainly 11Exx
• 15A66: Clifford algebras, spinors
• 15A69: Multilinear algebra, tensor products
• 15A72: Vector and tensor algebra, theory of invariants, See Also 13A50, 14D25
• 15A75: Exterior algebra, Grassmann algebras
• 15A78: Other algebras built from modules
• 15A90: Applications of matrix theory to physics
• 15A99: Miscellaneous topics

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Lütkepohl, H., "Handbook of matrices", John Wiley & Sons, Ltd., Chichester, 1996. ISBN 0-471-97015-8: no proofs, no algorithms, just the definitions and theorems.

The opposite extreme: Bourbaki, N., "Elements de mathématique: Algèbre", including "Chapitre 2: Algèbre linéaire", "Chapitre 3: Algèbre multilinéaire", and "Chapitre 9: Formes sesquilinéaires et formes quadratiques", all published by Hermann, Paris ca. 1958 (MR21 #6384, MR30#3104, MR27#5765)

An undergraduate text which opens up the post-matrix-computation perspective is by Axler, Sheldon: "Linear algebra done right", Springer-Verlag, New York, 1996. 238 pp. ISBN 0-387-94596-2; MR97i:15002

There is a substantial pool of information on numerical issues in linear algebra available on the internet -- see for example the newsgroup sci.math.num-analysis.

An online Elementary Linear Algebra text [Keith Matthews]

Software and tables

GAMS Linear algebra software

CLIFFORD, a Maple package for computations with Clifford algebras Cl(B) of any bilinear form B.

Macsyma's Atensor package (for tensor algebras) handles several kinds of tensor algebras, including universal tensor algebras, Grassmann, polynomial, Clifford, symplectic algebras. In contains simplifiers for these algebras. The developmental version of Macsyma includes (anti)derivation operators and, for Clifford algebras, an exponentiation operation.

Other web sites with this focus

• SIAM's Linear Algebra and Matrix Theory page.
• UTK archives page
• Here are the AMS and Goettingen resource pages for area 15.

Selected topics at this site

• Independence of the vector space axioms
• History of the Cauchy-Schwarz inequality
• Direct rotation -- shortest rotation taking one vector to another
• Proof of the Cayley-Hamilton Theorem
• Determinants, permanents, and immanents of a matrix
• Determinant-preserving endomorphisms of End(V)
• Using Dodgson's condensation formula to find determinants of symmetric Toeplitz matrices
• Inverses, determinants of Vandermonde-like special matrices (and the 'Advanced determinant calculus')
• Famous conjectures: existence of Hadamard matrices, projective planes, Jacobian conjecture
• What is the probability that two matrices will commute
• What are eigenvalues and linear transformations?
• How to compute eigenvectors (after the eigenvalues) for a 3x3 matrix.
• The determinant equals the product of the eigenvalues.
• Eigenvalues of a symmetric matrix and the symmetric part of a general square matrix.
• Multiple characterizations of positive (semi)definite matrices, and applications
• Why are eigenvalues of Hermitian matrices real?
• Gershgorian circles (for matrix eigenvalues)
• Eigenvalues of a circulant matrix.
• Generalized eigenvalue problem -- complete set of eigenvectors?
• Smallest matrix norms match largest eigenvalue
• The Spectral Radius Formula (for operator norm of matrices)
• Distribution of eigenvalues in random matrices
• Power method: successive Rayleigh quotients converge to dominant eigenvalue of a matrix
• Do the parts of the Jordan Decomposition of a matrix vary continuously with the matrix? (Not really)
• Is similarity achieved over the ground field? (yes)
• Example of companion matrices (to Chebyshev polynomials)
• Finding a symmetric matrix with prescribed characteristic polynomial
• When is a matrix irreducible (no invariant subspaces)?
• Generalized inverses of a matrix: definitions and applications
• The Moore-Penrose pseudo-inverse of a matrix.
• Computing the pseudoinverse using SVD.
• Using the pseudo-inverse and Tihonov regularization to solve linear systems of equations.
• Computing determinants of Toeplitz matrices
• Comparison of various factorizations of symmetric, positive definite matrices
• Some pointers on the computation of the Singular Value Decomposition of a matrix.
• Karhunen-Loève procedure: picks out the dominant terms of the Singular Value Decomposition (Proper Orthogonal Decomposition, Principal Component Analysis, analysis by Empirical Eigenfunctions)
• Lay person's description of Principal Component Analysis
• What is the Singular Value Decomposition
• Using the Cholesky factorization of a matrix to find an isometric embedding of a finite set of points.
• Not all symmetric matrices have a (modified) Cholesky-factorization
• Real Polar Decomposition of a real matrix as (symmetric positive semidefinite)*(orthogonal)
• Maple code to do QR decomposition of a matrix.
• Bunch-Parlett matrix decomposition (and counting negative eigenvalues)
• Every square real matrix is the product of two symmetric matrices
• Basic code for Gauss-Jordan inversion of a square matrix.
• Good algorithm for computing minimal polynomial?
• Matrix inversion by Monte-Carlo techniques(!): citations.
• Pointer to text on Matrix Algorithms [G B "Pete" Stewart]
• Efficient (recursive) methods of matrix multiplication (Strassen algorithm)
• Hints for solving a large linear system of equations.
• Solving a large sparse system of linear equations.
• Pointers to sparsity plots and other matrix software
• What are the multiplicative scalar functions on matrices? (determinants...)
• Counting annihilating matrices over a finite field.
• Use of permutation groups to determine a method for transposing nonsquare matrices in place.
• Counting the dimensions of magic squares and cubes.
• Example of expressing a vector as a linear combination of two others.
• Writing a matrix as a linear combination of orthogonal matrices.
• Finding linear combinations of matrices to have rank 1
• How to tell if a family of polynomials is linearly independent over Q.
• Given many vectors in a vector space, how to find linear relations among few of them?
• Definition and properties of the square-root function on (positive-definite) matrices
• General solution of matrix quadratic equations
• Lyapunov matrix equation, Sylvester matrix equation
• Origin and scope of the instability associated with the Hilbert matrix
• What is so ill-conditioned about the Hilbert matrix?
• Computing the inverse of the ill-conditioned Hilbert matrices
• Computing the determinant of the Hilbert matrices.
• Kantorovich's Inequality for positive definite Hermitian matrices.
• Matrix inequalities for positive definite matrices (Hadamard, Szász, etc.)
• Pointers to results on non-negative matrices (Perron-Frobenius theory)
• Extensions of Perron-Frobenius theorem to nonnegative matrices
• Vector spaces with periodic automorphism groups
• What is the tensor product of vector spaces?
• Application of exterior algebras to generalize the factorization of adj(X) when X is singular
• Current research trends in multilinear algebra
• Some references for multilinear algebra
• Application of Clifford algebras to quadratic forms
• Exploring Clifford algebras and Geometric Algebra and applications.
• Applications of Clifford algebras to differential topology and physics
• The game "Lights Out" is solved with (5 x 5) matrices over Z/2Z.