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Matrix
In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be
numbers or, more generally, any abstract quantities that can be added and multiplied. Matrices are used to
describe linear equations, keep track of the coefficients of linear transformations and to record data that
depend on multiple parameters. Matrices are described by the field of matrix theory. They can be added,
multiplied, and decomposed in various ways, which also makes them a key concept in the field of linear
algebra.
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 (written m × n) and m and n are called its dimensions. The
dimensions of a matrix are always given with the number of rows first, then the number of columns. It is
commonly said that an m-by-n matrix has an order of m × n ("order" meaning size). Two matrices of the
same order whose corresponding entries are equivalent are considered equal.
The entry that lies in the i-th row and the j-th column of a matrix is typically referred to as the i,j, or (i,j), or
(i,j)-th, or (i,j)th entry of the matrix. Again, the row is always noted first, then the column.
Almost always upper-case letters denote matrices, while the corresponding lower-case letters, with two
subscript indices, represent the entries. For example, the (i,j)th entry of a matrix A is most commonly
written as ai,j. Alternative notations for that entry are A[i,j] or Ai,j. In addition to using upper-case letters to
symbolize matrices, many authors use a special typographical style, commonly boldface upright (nonitalic), to further distinguish matrices from other variables. Following this convention, A is a matrix,
distinguished from A, a scalar. An alternate convention is to annotate matrices with their dimensions in
small type underneath the symbol, for example,
for an m-by-n matrix.
We often write
or
to define an m × n matrix
A. In this case, the entries ai,j are defined separately for all integers 1 ≤ i ≤ m and 1 ≤ j ≤ n. In some
programming languages, the numbering of rows and columns starts at zero. Texts which use any such
language extensively frequently follow that convention, so we have 0 ≤ i ≤ m-1 and 0 ≤ j ≤ n-1.
A matrix where one of the dimensions equals one is often called a vector, and interpreted as an element of
real coordinate space. An m × 1 matrix (one column and m rows) is called a column vector and a 1 × n
matrix (one row and n columns) is called a row vector.
An
matrix
is a function
where
is any non-empty set.
is the Cartesian product of sets
We say that matrix
is a matrix over the set
and
. Important thing to note is that, if we want to have matrix
algebra, the set
must be a ring and matrix
must be a square matrix. Since the set of all square
matrices over a ring is also a ring, matrix algebra is usually called matrix ring.
Since this article mainly considers matrices over real numbers, matrices shown here are actually functions
The matrix
or
is a
matrix. The element a2,3 or
is 7. In terms of the mathematical definition given above,
this matrix is a function
and, for example,
and
The matrix
is a
matrix, or 9-element row vector.
Two or more matrices of identical dimensions m and n can be added. Given m-by-n matrices A and B, their
sum A+B is the m-by-n matrix computed by adding corresponding elements:
For example:
Another, much less often used notion of matrix addition is the direct sum.
Scalar multiplication
Given a matrix A and a number c, the scalar multiplication cA is computed by multiplying every element of
A by the scalar c (i.e.
). For example:
Matrix addition and scalar multiplication turn the set
with real entries into a real vector space of dimension
of all m-by-n matrices
.
Matrix multiplication
Multiplication of two matrices is well-defined only if the number of columns of the left matrix is the same
as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their
matrix product AB is the m-by-p matrix given by:
for each pair (i,j). For example:
Matrix multiplication has the following properties:
 (AB)C = A(BC) for all k-by-m matrices A, m-by-n matrices B and n-by-p matrices C
("associativity").
 (A+B)C = AC+BC for all m-by-n matrices A and B and n-by-k matrices C ("right distributivity").
 C(A+B) = CA+CB for all m-by-n matrices A and B and k-by-m matrices C ("left distributivity").
Matrix multiplication is not commutative; that is, given matrices A and B and their product defined, then
generally AB
BA. It may also happen that AB is defined but BA is not defined.
Besides the ordinary matrix multiplication just described, there exist other operations on matrices that can
be considered forms of multiplication, such as the Hadamard product and the Kronecker product.