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matrix – natural number
matrix
A rectangular array of, usually, real or complex numbers, organized into rows and columns. When specifying the size of a matrix, the number of rows is stated first. For example, here is a 2 by 4 real matrix:
The numbers in a matrix are called entries, and are specified by the row and then column in which they appear. In the example above, for instance, A1,2 is the entry in the first row and second column, i.e. 4.98.
 Two matrices of the same size (with the same number of rows and columns) can be added componentwise, that is (A + B)i, j = Ai, j + Bi, j. Matrix multiplication is a little more complicated. A matrix M can be multiplied on the right by another matrix T only if T has the same number of rows as M has columns, and the result will have as many rows as M and as many columns as T. The entry of M T appearing in row i and column j is the sum of the pairwise products of the entries in row i of M and column j of T. In other words, if M is an n by m matrix, T must be an m by l matrix, and
Notice that this operation is not, in general, commutative.
A matrix that has the same number of rows and columns is called a square matrix, and these are particularly interesting as they can be added and multiplied, and the results will have the same dimensions.
The most standard use of matrices is to represent linear operators, but they have a wide variety of other uses, and are interesting to study in their own right. Matrices can have a large number of different types of entries, including various kinds of numbers, elements of abstract fields and other algebraic structures, functions, and so forth.
measurable function
Given measure spaces (X, M, m) and (Y, N, n), a function f from X to Y is said to be measurable if the inverse image of every measurable set in Y is measurable in X.
measurable space
A set X together with a s-algebra of sets M defined on it is called a measurable space, usually denoted (X, M). The elements of M are called measurable sets.
Cf. measure.
measure
Given a set X and a s-algebra of sets M defined on X, a measure on X is a positive real-valued function m whose domain is M, and which satisfies: - the measure of the empty set is zero.
- (Countable additivity) For any countable, disjoint sequence of sets in the domain of m, the measure of the union of the sequence is equal to the sum of the measures of the sets in the sequence.
If the measure of every set in M is finite, then m is called a finite measure. If X can be expressed as a union of sets in M all of whose measures are finite, then m is called a s-finite measure. If for every set E in the domain of m whose measure is infinite there is a subset F of E which has finite measure, then m is called a semi-finite measure. If for any set E in M which has zero measure all the subsets of E are also in M (and therefore have zero measure), then m is called a complete measure.
Cf. measure space, signed measure, outer measure, Lebesgue measure, Borel measure.
measure space
A set X together with a s-algebra of sets M defined on it and a measure m defined on M is called a measure space, usually denoted by (X, M, m).
meet
A binary operation whose value on two elements a and b of a lattice is the greatest lower bound of a and b, denoted a  b.
Cf. join.
metric
Given a set X, a metric on X is a function d with domain X and range in R* (the set of non-negative real numbers) satisfying, for all a, b, and c in X:- d(a,b) = 0 if and only if a = b;
- d(a,b) = d(b,a); and
- d(a,b) + d(b,c) >= d(a,c
(where >= means ‘greater than or equal to’).
Cf. metric space.
metric space
A set with a metric defined on its elements.
module
A generalization of the notion of vector space. Specifically, if M is an Abelian group under addition, and R is a ring such that rm is in M whenever m is an element of M and r is an element of R, then M is called a left R-module (a right R-module is defined analogously), provided- r(m + n) = rm + rn,
- (r + s)m = rm + sm, and
- r(sm) = (rs)m
for all r, s in R and m, n in M. A module is cyclic if there is a generating element m of M such that every element of M is of the form rm for some r in R. A module is finitely generated if there are elements m1, m2, ..., mk in M such that every element of M is of the form r1m1 + r2m2 + ... + rkmk for some r1, r2, ..., rk in R.
monoid
See group.
monomorphism
A morphism f from X to Y is called a monomorphism when it is injective, that is, when to each element y of Y there corresponds at most one x in X such that f(x) = y.
Cf. epimorphism.
monotone function
Also called monotonic function. See order-preserving function.
morphism
A function from one set to another is called a morphism if it preserves some designated structural properties or operations on the domain set. Typically, the word morphism is not used by itself, but in combination with a prefix that indicates whether it is injective, surjective, etc.
Cf. automorphism, epimorphism, homeomorphsim, homomorphism, isomorphism, monomorphsim.
multiplication
A binary operation on numbers or quantities resulting in a product, usually but not always amounting to repeated addition. On the natural numbers multiplication is defined recursively by the Peano axioms, such that the product of two numbers n and m, denoted by n × m, is found by adding up m copies of n (or n copies of m).
Multiplication of most kinds of numbers is associative and commutative, but these properties sometimes fail, for example in the case of matrix multiplication.
When a product of more than two numbers or quantities is taken, the general product may be denoted by the capital Greek letter Pi, i.e., Pai denotes the product a1 × a2 × . . . × an.
multiply
To find the product of two numbers or quantities by multiplication.
mutually prime
Two integers are mutually prime if they have no common factors larger than 1 or -1.
mutually singular
Given a measurable space (X, M), two signed measures m and n on X are called mutually singular, denoted by
if there exist sets E and F in M such that their union is X, their intersection is empty, and m(E) = 0 and n(F) = 0.
nth-term test
A test for the divergence of a series. See the related article for a complete description.
Related article: Series
natural base
See Euler number.
natural logarithm
A logarithm with base e, the Euler number. Often written “ln” rather than “log” to distinguish it from logarithms using other bases.
natural number
An element of the set N = {1, 2, 3, ...} consisting of all the “counting numbers.” When the number 0 is included, this set is sometimes called the whole numbers. In set theory, the natural numbers (incuding 0) are identified with the set w of finite ordinals. The natural numbers are a well-founded linear order with no largest member, and are countably infinite.
Cf. Peano axioms, rational number, real number.
Related MiniText: Number -- What Is How Many?
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