- Matrix theory
Matrix theory is a branch of
mathematicswhich focuses on the study of matrices. Initially a sub-branch of linear algebra, it has grown to cover subjects related to graph theory, algebra, combinatorics, and statisticsas well.
The term "matrix" was first coined in 1848 by
J.J. Sylvesteras a name of an array of numbers. In 1855, Arthur Cayleyintroduced matrix as a representation of linear transformation. This period was considered as the beginning of linear algebraand matrix theory. The motivation for linear algebra, and the first use of matrices, was the study of systems of linear equations. Related concepts such as determinantand Gaussian elimination, which existed long before the introduction of matrices, are now part of matrix theory.
The study of
vector spaceover finite field, a branch of linear algebra which is useful in coding theory, naturally leads to the study and use of matrices over finite field in coding theory.
Modules are generalizations of vector spaces. They are similar to vector spaces, but defined over rings rather than fields. This leads to the study of matrices over rings. Matrix theory in this area is not often considered as a branch of linear algebra. Among the results listed in Useful theorems, the
Cayley-Hamilton Theoremis valid if the underlying ring is commutative, Smith normal formis valid if the underlying ring is a principal ideal domain, but others are valid for only matrices over complex numbers or real numbers. Magic squares and Latin squares, two ancient branches of recreational mathematics, are now reformulated using the language of matrices. The link between Latin squares and coding theory demonstrates that this is not merely a coincidence.
With the advance of
computertechnology, it is now possible to solve systems of large numbers of linear equations in practice, not just in theory. John von Neumannand Herman Goldstineintroduced condition numbers in analyzing round-off errors in 1947. Later, different techniques to calculation, multiplication or factorization of matrices were invented, such as the Fast Fourier Transform.
payoff matrixin game theory, also introduced by John von Neumann, might be the first application of matrices to economics.
simplex algorithm, a technique involving the operations of matrices of very large size, is used to solve operations researchproblems, a field strongly related to economics. Flow networkproblems, part of both graph theoryand linear programming, can be solved using the simplex algorithm, although there are other more efficient methods. Matrices appear elsewhere in graph theory as well; for example, the adjacency matrixrepresentation of a directed or undirected graph. Important matrices in combinatoricsare permutation matrices, which represent permutations, and Hadamard matrices.
Both adjacency matrices of graphs and permutation matrices are examples of nonnegative matrices, which also include stochastic and doubly stochastic matrices. Stochastic matrices are useful in the study of
stochastic processes, in probability theoryand in statistics. The evaluation of an enormous stochastic matrix is the central idea behind the PageRankalgorithm used by convex combinationof permutation matrices.
Another important tool in statistics is the
For optimization problems involving multi-variable real-value functions, Positive-definite matrices occur in the search for
maxima and minima.
There are also practical uses for matrices over arbitrary rings (see
Matrix ring). In particular, matrices over polynomial rings are used in control theory.
On the pure mathematics side, matrix rings can provide a rich field of counterexamples for mathematical conjectures, amongst other uses. The square matrices also plays a special role, because the "n"×"n" matrices for fixed "n" have many closure properties.
* Jordan decomposition
Singular value decomposition
Smith normal form
List of matrices. This list is a rich source of information and links to a very wide variety of matrices from mathematics, science and engineering.
Real matrices (2 x 2)shows that, when non-singular, a 2 x 2 real matrix is proportional to a shear mapping, a squeeze mapping, or a rotation.
* Beezer, Rob, [http://linear.ups.edu/index.html "A First Course in Linear Algebra"] , licensed under GFDL.
* Jim Hefferon: " [http://joshua.smcvt.edu/linalg.html/ Linear Algebra] " (Online textbook)
* [http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html A Brief History of Linear Algebra and Matrix Theory]
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