Matrix (mathematics) |
For the square matrix section, see Matrix (mathematics)#Square matrices and related definitions.
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a Ring (mathematics)-like algebraic structure. In this article, the entries of a matrix are real number or complex number numbers unless otherwise noted.
Matrices are useful to record data that depend on two categories, and to keep track of the coefficients of system of linear equations and linear transformations.
For the development and applications of matrices, see matrix theory.
= Definitions and notations =
The horizontal lines in a matrix are called rows and the vertical lines are called columns. A matrix with m rows and n columns is called an m -by- n matrix (or m Ã? n matrix) and m and n are called its dimensions.
The entry of a matrix A that lies in the i -th row and the j -th column is called the i,j entry or ( i , j )-th entry of A . This is written as A i,j or A [ i,j ].
We often write A:=(a_{i,j})_{m imes n} to define an m Ã? n matrix A with each entry in the matrix A [ i,j ] called a ij for all 1 i m and 1 j n .
== Example ==
The matrix
:egin{bmatrix} 1 & 2 & 3 \ 1 & 2 & 7 \ 4&9&2 \ 6&0&5end{bmatrix}
is a 4Ã?3 matrix. The element A [2,3] or a 2,3 is 7.
= Adding, subtracting, and multiplying matrices =
==Sum==
If two m -by- n matrices A and B are given, we may define their sum A + B as the m -by- n matrix computed by adding corresponding elements, i.e., ( A + B )[ i, j ] = A [ i, j ] + B [ i, j ]. For example
: egin{bmatrix} 1 & 3 & 2 \ 1 & 0 & 0 \ 1 & 2 & 2 end{bmatrix} + egin{bmatrix} 0 & 0 & 5 \ 7 & 5 & 0 \ 2 & 1 & 1 end{bmatrix} = egin{bmatrix} 1+0 & 3+0 & 2+5 \ 1+7 & 0+5 & 0+0 \ 1+2 & 2+1 & 2+1 end{bmatrix} = egin{bmatrix} 1 & 3 & 7 \ 8 & 5 & 0 \ 3 & 3 & 3 end{bmatrix}
Another, much less often used notion of matrix addition is the direct sum (matrix).
==Difference==
==Scalar multiplication==
If a matrix A and a number c are given, we may define the scalar multiplication cA by ( cA )[ i , j ] = cA [ i , j ]. For example
:2 egin{bmatrix} 1 & 8 & -3 \ 4 & -2 & 5 end{bmatrix} = egin{bmatrix} 2 imes 1 & 2 imes 8 & 2 imes -3 \ 2 imes 4 & 2 imes -2 & 2 imes 5 end{bmatrix} = egin{bmatrix} 2 & 16 & -6 \ 8 & -4 & 10 end{bmatrix}
These two operations turn the set M( m , n , R) of all m -by- n matrices with real entries into a real vector space of dimension mn .
==Multiplication==
Multiplication of two matrices is well-defined only if the number of columns of the first matrix is the same as the number of rows of the second matrix. If A is an m -by- n matrix ( m rows, n columns) and B is an n -by- p matrix ( n rows, p columns), then their product AB is the m -by- p matrix ( m rows, p columns) given by
:( AB )[ i , j ] = A [ i , 1] * B [1, j ] + A [ i , 2] * B [2, j ] + ... + A [ i , n ] * B [ n , j ] for each pair i and j . For instance: : egin{bmatrix} 1 & 0 & 2 \ -1 & 3 & 1 \ end{bmatrix} imes egin{bmatrix} 3 & 1 \ 2 & 1 \ 1 & 0 end{bmatrix} = egin{bmatrix} (1 imes 3 + 0 imes 2 + 2 imes 1) & (1 imes 1 + 0 imes 1 + 2 imes 0) \ (-1 imes 3 + 3 imes 2 + 1 imes 1) & (-1 imes 1 + 3 imes 1 + 1 imes 0) \ end{bmatrix} = egin{bmatrix} 5 & 1 \ 4 & 2 \ end{bmatrix}
This multiplication has the following properties:
It is important to note that commutativity does not generally hold; that is, given matrices A and B and their product defined, then generally AB BA.
Matrices are said to anticommute if AB = -BA. Such matrices are very important in representations of Lie algebras and in Representations of Clifford algebras
= Linear transformations, ranks and transpose =
Matrices can conveniently represent linear transformation because matrix multiplication neatly corresponds to the composition of maps, as will be described next. This same property makes them powerful data structures in high-level programming languages.
Here and in the sequel we identify R n with the set of rows or n -by-1 matrices. For every linear map f : R n -> R m there exists a unique m -by- n matrix A such that f ( x ) = Ax for all x in R n . We say that the matrix A represents the linear map f . Now if the k -by- m matrix B represents another linear map g : R m -> R k , then the linear map g o f is represented by BA . This follows from the above-mentioned associativity of matrix multiplication.
More generally, a linear map from an n -dimensional vector space to an m -dimensional vector space is represented by an m -by- n matrix, provided that Basis (linear algebra) have been chosen for each.
The rank of a matrix of a matrix A is the Hamel dimension of the Image (mathematics) of the linear map represented by A ; this is the same as the dimension of the space generated by the rows of A , and also the same as the dimension of the space generated by the columns of A .
The transpose of an m -by- n matrix A is the n -by- m matrix A tr (also sometimes written as A T or t A ) formed by turning rows into columns and columns into rows, i.e. A tr[ i , j ] = A [ j , i ] for all indices i and j . If A describes a linear map with respect to two bases, then the matrix A tr describes the transpose of the linear map with respect to the dual bases, see dual space.
We have ( A + B )tr = A tr + B tr and ( AB )tr = B tr * A tr.
= Square matrices and related definitions =
A square matrix is a matrix which has the same number of rows as columns. The set of all square n -by- n matrices, together with matrix addition and matrix multiplication is a ring (mathematics). Unless n = 1, this ring is not commutative ring.
M( n , R), the ring of real square matrices, is a real unitary associative algebra. M( n , C), the ring of complex square matrices, is a complex associative algebra.
The unit matrix or identity matrix In , with elements on the main diagonal set to 1 and all other elements set to 0, satisfies MIn=M and InN=N for any m -by- n matrix M and n -by- k matrix N . For example, if n = 3: : I_3 = egin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 end{bmatrix}
The identity matrix is the identity element in the ring of square matrices.
Invertible elements in this ring are called invertible matrix or non-singular matrices. An n by n matrix A is invertible if and only if there exists a matrix B such that : AB = I n ( = BA ). In this case, B is the inverse matrix of A , denoted by A −1. The set of all invertible n -by- n matrices forms a group (mathematics) (specifically a Lie group) under matrix multiplication, the general linear group.
If is a number and v is a non-zero vector such that A v = v, then we call v an eigenvector of A and the associated eigenvalue. (Eigen means own in German.) The number is an eigenvalue of A if and only if A − I n is not invertible, which happens if and only if p A () = 0. Here p A ( x ) is the characteristic polynomial of A . This is a polynomial of degree n and has therefore n complex roots (counting multiple roots according to their multiplicity). In this sense, every square matrix has n complex eigenvalues.
The Determinant of a square matrix A is the product of its n eigenvalues, but it can also be defined by the Determinant#General definition and computation . Invertible matrices are precisely those matrices with nonzero determinant.
The .
The trace of a matrix of a square matrix is the sum of its diagonal entries, which equals the sum of its n eigenvalues.
Every orthogonal matrix is a square matrix.
Matrix exponential is defined for square matrices, using power series.
= Special types of matrices =
In many areas in mathematics, matrices with certain structure arise. A few important examples are
For a more extensive list see list of matrices.
= Matrices in abstract algebra =
If we start with a Ring (mathematics) R , we can consider the set M( m , n , R ) of all m by n matrices with entries in R . Addition and multiplication of these matrices can be defined as in the case of real or complex matrices (see #Adding and multiplying matrices). The set M( n , R ) of all square n by n matrices over R is a ring in its own right, isomorphic to the endomorphism ring of the left R -module (mathematics) R n .
Similarly, if the entries are taken from a semiring S , matrix addition and multiplication can still be defined as usual. The set of all square n Ã? n matrices over S is itself a semiring. Note that fast matrix multiplication algorithms such as the Strassen algorithm generally only apply to matrices over rings and will not work for matrices over semirings that are not rings.
If R is a commutative ring, then M( n , R ) is a unitary associative algebra over R . It is then also meaningful to define the Determinant of square matrices using the Leibniz formula ; a matrix is invertible if and only if its determinant is invertible in R .
All statements mentioned in this articles for real or complex matrices remain correct for matrices over an arbitrary field (mathematics).
Matrices over a polynomial ring are important in the study of control theory.
=History=
The study of matrices is quite old. Latin squares and magic squares have been studied since prehistoric times.
Matrices have a long history of application in solving linear equations. Gottfried Leibniz, one of the two founders of calculus, developed the theory of Determinants in 1693. Gabriel Cramer developed the theory further, presenting Cramer s rule in 1750. Carl Friedrich Gauss and Wilhelm Jordan developed Gauss-Jordan elimination in the 1800s.
The term matrix was first coined in 1848 by J. J. Sylvester. George Cayley, William Rowan Hamilton, Hermann Grassmann, Ferdinand Georg Frobenius and John von Neumann are among the famous mathematicians who have worked on matrix theory.
Olga Taussky Todd (1906-1995) started to use matrix theory when investigating an aerodynamic phenomenon called flutter, during WWII.
= Further reading =
A more advanced article on matrices is matrix theory.
= External links =
*[http://www.maths.hscripts.com/matrix/index.php Matrix Calculators: dynamic online calculators] *[http://www.ualr.edu/~lasmoller/matrices.html Matrix name and history: very brief overview] *[http://wims.unice.fr/wims/wims.cgimodule=tool/linear/matrix.en WIMS Matrix Calculator] computes determinant, rank, inverse etc. online.|
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