matrix

ma·trix

M0153500 (mā′trĭks)n. pl. ma·tri·ces (mā′trĭ-sēz′, măt′rĭ-) or ma·trix·es 1. A situation or surrounding substance within which something else originates, develops, or is contained: "Freedom of expression is the matrix, the indispensable condition, of nearly every form of freedom" (Benjamin N. Cardozo).2. The womb.3. Anatomy a. The formative cells or tissue of a specialized structure such as a hair, nail, claw, or tooth.b. See ground substance.4. Geology a. The solid matter in which a fossil or crystal is embedded.b. Groundmass.5. A mold or die.6. The principal metal in an alloy, as the iron in steel.7. A binding substance, as cement in concrete.8. a. Mathematics A rectangular array of numeric or algebraic quantities subject to mathematical operations.b. Something resembling such an array, as in the regular formation of elements into columns and rows.9. Computers The network of intersections between input and output leads in a computer, functioning as an encoder or a decoder.10. Printing a. A mold used in stereotyping and designed to receive positive impressions of type or illustrations from which metal plates can be cast. Also called mat².b. A metal plate used for casting typefaces.11. An electroplated impression of a phonograph record used to make duplicate records.

[Middle English matrice, from Old French, from Late Latin mātrīx, mātrīc-, from Latin, breeding-animal, from māter, mātr-, mother; see māter- in Indo-European roots.]

matrix

(ˈmeɪtrɪks; ˈmæ-) n, pl matrices (ˈmeɪtrɪˌsiːz; ˈmæ-) or matrixes1. a substance, situation, or environment in which something has its origin, takes form, or is enclosed2. (Anatomy) anatomy the thick tissue at the base of a nail from which a fingernail or toenail develops3. (Biology) the intercellular substance of bone, cartilage, connective tissue, etc4. (Geological Science) a. the rock material in which fossils, pebbles, etc, are embeddedb. the material in which a mineral is embedded; gangue5. (Printing, Lithography & Bookbinding) printing a. a metal mould for casting typeb. a papier-mâché or plastic mould impressed from the forme and used for stereotyping. Sometimes shortened to: mat 6. (Electronics) (formerly) a mould used in the production of gramophone records. It is obtained by electrodeposition onto the master7. (Mechanical Engineering) a bed of perforated material placed beneath a workpiece in a press or stamping machine against which the punch operates8. (Metallurgy) metallurgy a. the shaped cathode used in electroformingb. the metal constituting the major part of an alloyc. the soft metal in a plain bearing in which the hard particles of surface metal are embedded9. (Chemistry) the main component of a composite material, such as the plastic in a fibre-reinforced plastic10. (Mathematics) maths a rectangular array of elements set out in rows and columns, used to facilitate the solution of problems, such as the transformation of coordinates. Usually indicated by parentheses: (

). Compare determinant³11. (Linguistics) linguistics the main clause of a complex sentence12. (Computer Science) computing a rectangular array of circuit elements usually used to generate one set of signals from another13. (Anatomy) obsolete the womb[C16: from Latin: womb, female animal used for breeding, from māter mother]

ma•trix

(ˈmeɪ trɪks, ˈmæ-)

n., pl. ma•tri•ces (ˈmeɪ trɪˌsiz, ˈmæ-) ma•trix•es. 1. something that constitutes the place or point from which something else originates. 2. a formative tissue, as the epithelium from which nails grow. 3. a. the intercellular substance of a tissue. b. ground substance (def. 1). 4. the fine-grained portion of a rock in which coarser crystals or rock fragments are embedded. 5. gangue. 6. a crystalline phase in an alloy in which other phases are embedded. 7. a mold for casting typefaces. 8. (in a press or stamping machine) a multiple die or perforated block on which the material to be formed is placed. 9. a rectangular array of numbers, algebraic symbols, or mathematical functions, esp. when such arrays are added and multiplied according to certain rules. 10. a similar rectangular array consisting of rows and columns of numbers, symbols, etc., used in displaying statistical variables, linguistic features, or other data. [1325–75; < Latin mātrīx female animal kept for breeding (Late Latin: register, orig. of such beasts), parent stem (of plants), derivative of māter mother]

ma·trix

(mā′trĭks) A substance within which something is contained or embedded. The mineral grains of a rock in which fossils are embedded make up a matrix. Bone cells are embedded in a matrix of collagen fibers and mineral salts.

matrix

- From Latin, meaning "breeding female," it originally was used for the uterus or womb, then for a supporting or enclosing structure.See also related terms for womb.

Thesaurus

Noun	1.	matrix - (mathematics) a rectangular array of quantities or expressions set out by rows and columns; treated as a single element and manipulated according to rulesmath, mathematics, maths - a science (or group of related sciences) dealing with the logic of quantity and shape and arrangementcorrelation matrix - a matrix giving the correlations between all pairs of data setsarray - an orderly arrangement; "an array of troops in battle order"dot matrix - a rectangular matrix of dots from which written characters can be formedsquare matrix - a matrix with the same number of rows and columnsreal matrix - a matrix whose elements are all real numberstranspose - a matrix formed by interchanging the rows and columns of a given matrix
	2.	matrix - (geology) amass of fine-grained rock in which fossils, crystals, or gems are embeddedgeology - a science that deals with the history of the earth as recorded in rocksrock, stone - material consisting of the aggregate of minerals like those making up the Earth's crust; "that mountain is solid rock"; "stone is abundant in New England and there are many quarries"
	3.	matrix - an enclosure within which something originates or develops (from the Latin for womb)natural enclosure, enclosure - a naturally enclosed space
	4.	matrix - the body substance in which tissue cells are embeddedground substance, intercellular substancebody substance - the substance of the bodybone, os - rigid connective tissue that makes up the skeleton of vertebratesconnective tissue - tissue of mesodermal origin consisting of e.g. collagen fibroblasts and fatty cells; supports organs and fills spaces between them and forms tendons and ligamentscartilage, gristle - tough elastic tissue; mostly converted to bone in adults
	5.	matrix - the formative tissue at the base of a nailanimal tissue - the tissue in the bodies of animalsnail - horny plate covering and protecting part of the dorsal surface of the digits
	6.	matrix - mold used in the production of phonograph records, type, or other relief surfacecast, mold, mould - container into which liquid is poured to create a given shape when it hardens

matrix

nounA hollow device for shaping a fluid or plastic substance:cast, form, mold.

Translations

Matrix

matrix,

in mathematics, a rectangular array of elements (e.g., numbers) considered as a single entity. A matrix is distinguished by the number of rows and columns it contains. The matrix

is a 2×3 (read "2 by 3") matrix, because it contains 2 rows and 3 columns. A matrix having the same number of rows as columns is called a square matrix. The matrix

is a 2×2 matrix, or square matrix of order 2; a square matrix of order n contains n rows and n columns. Definitions are made for certain operations with matrices; for example, a matrix may be multiplied by a number, and two matrices of the same order may be added or multiplied using an algebra of matrices that has been developed. Matrices find application in such fields as vector analysis and the solution of systems of linear equations by means of electronic computers.

Bibliography

See R. C. Dorfi, Matrix Algebra (1969).

Matrix

in cytology, the homogeneous fine-grained matter that fills intracellular structures (organoids) and the spaces between them.

There are three types of matrix: cytoplasmic matrix, which, depending on the physiological state of the cell, is capable of viscid flow or elastic deformation; mitochondrial matrix, the semifluid matter that fills the spaces between the cristae, or crests, of the mitochondria; and the matrix of the nucleus, plastids, and other organoids. Cytoplasmic matrix consists chiefly of protein molecules aggregated to various degrees and serving as a supportive medium for the cellular organoids. It holds basal bodies, centrioles, filaments, microtubules, and other fibrillar structures whose functions have not been completely elucidated.

REFERENCE

Frey-Wyssling, A., and K. Mühlthaler. Ul’trastruktura rastitel’noi kletki. Moscow, 1968. (Translated from English.)
Loewy, A., and F. Siekevitz. Struktura i funktsii kletki. Moscow, 1971. (Translated from English.)

Matrix

in mathematics, a system of elements a_ij (numbers, functions, or other quantities for which there are defined algebraic operations) arranged in the form of a rectangular array. If the array has m rows and n columns, we speak of an (m × n) matrix (m by n matrix). It is denoted by

and in abbreviated notation, by ║a_ij║ or (a_ij). Both finite matrices and matrices with an infinite number of rows or columns are considered.

A matrix consisting of a single row is called a row matrix, and that consisting of a single column is called a column matrix. If m = n, the matrix is called a square matrix of order n. A square matrix in which only the diagonal elements α = α_ii are nonzero is called a diagonal matrix and is denoted by diag (α₁, …, α_n). If all α_i = a, we speak of a scalar matrix. When α = 1, the matrix is called an identity matrix and is denoted by E. A matrix all of whose elements are zero is called a zero matrix.

By interchanging the rows and columns in a given matrix A, we obtain the transpose A′, or A^T, of A. If we replace the elements of a matrix by their complex conjugates, we obtain the complex conjugate matrix Ā of A. If we replace the elements of the transpose A′ of Ā by their complex conjugates, we obtain the matrix A*, called the conjugate transpose of A. The determinant of a square matrix A is denoted by │A│, or det A. A determinant of kth order consisting of elements at the intersection of some k rows and k columns of the matrix A in their natural arrangement is called a minor of kth order of the matrix. The rank of a matrix is the maximal order of the nonzero minors of the matrix.

Operations on matrices. The product of a rectangular (m × n) matrix A and the number α is the matrix whose elements are obtained by multiplying every element a_ij of A by α:

Addition is defined for identically structured rectangular matrices, and the elements of the sum are the sums of the corresponding elements of the two summands, that is,

Multiplication of matrices is defined only for rectangular matrices in which the number of columns of the first factor is equal to the number of rows of the second. The product of the (m × p) matrix A and the (p × n) matrix B is the (m × n) matrix C with elements

c_ij = a_i1b_1j + a_i2b_2j + … + a_ipb_pj

i = 1, …, m j = 1, …, n

These three operations on matrices possess properties similar to those of the operations on numbers. The exception is the noncommutativity of matrix multiplication, which means that the equality AB = BA may not hold. If AB = BA, then the matrices A and B are said to commute. Moreover, the product of two matrices may be the zero matrix even though neither factor is zero. The following rules hold:

The determinant of the product of two square matrices is equal to the product of their determinants.

It is often convenient to divide a matrix into blocks, each of which constitutes a matrix of lesser dimension, by drawing a line through the entire matrix from left to right or from top to bottom. When multiplying such a block matrix by a number, it is necessary to multiply each block by that same number. The addition and multiplication of matched block matrices are carried out as if the blocks were actully numbers.

The square matrix A = (a_ij) is nonsingular if its determinant is not zero; otherwise the matrix is singular. A matrix A⁻¹ is an inverse of the square matrix A if AA⁻¹ = E in this case, a_jk⁽⁻¹⁾ = A_ki/│A│, where A_ki denotes the cofactor of a_ki,. The nonsingularity of a matrix A is a necessary and sufficient condition for the existence of an inverse of A. If A has an inverse, then that inverse is necessarily unique and commutes with A. We have

(AB)⁻¹ = B⁻¹A⁻¹

Of great interest is the generalized inverse (or pseudoinverse) matrix A⁺, defined both for any rectangular matrix and for a singular square matrix. This matrix is defined by the four equalities

AA⁺A = A A⁺AA⁺ = A

AA⁺ = (AA⁺)* A⁺A = (A⁺A)*

Square matrices. The nth power Aⁿ of the matrix A is the product of n factors equal to A. An expression of the form α₀Aⁿ + α₁Aⁿ⁻¹ + … + α_nE, where α₀, α₁, …, α_n are numbers, is called the value of the polynomial α₀tⁿ + α₁tⁿ⁻¹ + … + α_nE at (the square matrix) A. The rules of operation on polynomials in a given matrix A are identical to the rules of operation on algebraic polynomials. We can also consider analytic functions of a matrix. In particular, if

is a series converging in the entire complex plane (for example, if f(t) = e^t) then the infinite series

is a convergent series for any matrix A and it is natural to put its sum equal to f(A). But if f(t) has a finite circle of convergence, then f(A) is defined by this series for “sufficiently small” matrices.

Analytic functions of a matrix play a major role in the theory of differential equations. Thus, a system of ordinary differential equations with constant coefficients, written in matrix notation in the form

dx/dt = AX

(here, X is the column of unknown functions), has the solution x = e^AtC, where C is a column of arbitrary constants.

A nonzero column vector X such that AX = λX is called an eigenvector of the matrix A. In this equality, the coefficient λ can only be one of the roots of the polynomial

which is called the characteristic polynomial of the matrix A. These roots are called the eigenvalues, or characteristic numbers, of A. The coefficients of the characteristic polynomial can be expressed in terms of sums of certain minors of A. In particular, p₁ = a₁₁ + • • • + a¹ⁿ = Tr A (trace of A) and p_n = (−1)ⁿ⁻¹│A│ We have the Cay ley-Hamilton theorem: if 0(t) is the characteristic polynomial of the matrix A, then 0(A) = 0, so that A is a “root” of its characteristic polynomial.

A matrix A is said to be similar to a matrix B if there exists a nonsingular matrix C such that B = C⁻¹AC. It can be easily verified that similar matrices have identical characteristic polynomials.

Matrix calculus. Matrices are useful tools for investigating many problems in theoretical and applied mathematics. One of the most important problems is that of finding solutions of systems of linear algebraic equations. In matrix notation these equations are written in the form

AX = F

where A is the coefficient matrix; X is the desired solution, written in the form of a column of n elements; and F is the column of free terms consisting of m elements. If A is a square nonsingular matrix, then the system has the unique solution X = A⁻¹F. If A is a rectangular (m × n) matrix of rank k, then the solution may not exist or may not be unique. When no solution exists, a generalized solution giving a minimum of the sum of the squares of the discrepancies may prove useful. When neither the exact nor generalized solution is unique, a normal solution, that is, a solution with the least sum of the squares of the components, is selected. The normal generalized solution is found from the formula X = A⁺F. The most important case is that of an overdetermined system k = n < m. In that case, the generalized solution is unique. When k = m < n (underdetermined system), then there exist infinitely many exact solutions and the formula above yields the normal solution.

No less important for many applications (in the theory of differential equations, the theory of small vibrations, quantum mechanics) is the solution of the complete or partial eigenvalue problem, which consists in finding all or some of the eigenvalues of a matrix and the corresponding eigenvectors or root vectors (certain generalized eigenvectors). This problem is closely related to the generalized eigenvalue problem, in which numbers and vectors are found such that AX = λBX (A and B are given matrices), and to many related problems.

The problem of reducing a square matrix to canonical form by means of similarity transformations is also directly connected with the complete eigenvalue problem. This form will be diag (λ₁, …, λ_n,) if the matrix has n different eigenvalues λ₁, …, λ_n,, or a Jordan form in the general case. (The Jordan canonical form of a square matrix is a matrix whose principal diagonal consists of the eigenvalues of the original matrix and whose diagonal just above or just below the principal diagonal consists of ones and zeros. The remaining entries are all equal to zero.)

Because of the great practical importance of these problems, there exist many different methods for their numerical solution. In addition to finding a numerical solution, it is also important to evaluate the “quality” of the solution found and to investigate the stability of the problem to be solved.

Special types of matrices. There exist numerous types of matrices, depending on the relationships between the elements. Some types arise naturally in various applications. Table 1 gives some of the most important types of square matrices.

Table 1. Types of matrices
Matrix	Defining condition
Symmetric	A = A′
Skew symmetric	A = −A′
Orthogonal	AA′ = E or A⁻¹ = A′
Stochastic
Hermitian	A = A*
Unitary	AA = E* or A⁻¹ = A*

We also note quasidiagonal matrices, that is, matrices whose nonzero elements are on the principal diagonal and on the diagonals on either side of the principal diagonal, for example, 2-diagonal and 3-diagonal matrices.

No less important are special types of matrices that are used as auxiliary matrices. These are the elementary matrices, which differ from the identity matrix in a single element, and matrices of rotation and reflection.

Some additional types of matrices are unitary analogs of rotation and reflection matrices; upper (lower) triangular matrices, in which the elements below (above) the principal diagonal are equal to zero; and upper (lower) almost-triangular matrices (Hessenberg-type matrices), in which the elements below (above) the next diagonal below (above) the principal diagonal are equal to zero.

Matrix transformation. Numerical methods of solving systems of linear equations are usually based on the transformation of the systems by a series of multiplications on the left by suitable auxiliary matrices in order to obtain an easily solvable system. Elementary matrices, matrices of rotation, or matrices of reflection are used as auxiliary matrices for real matrices. A system with a nonsingular matrix reduces either to a system with a triangular matrix or to one with an orthogonal matrix. Theoretically this is equivalent to the representation of the coefficient matrix in the form of a product of two triangular matrices (under certain additional conditions) or in the form of a product of a triangular matrix and an orthogonal matrix in either order.

For an overdetermined system it is possible to arrive at a system with a triangular matrix of order n, whose solution yields a generalized solution of the initial system, by multiplying on the left by a chain of matrices of rotation or reflection.

To solve an eigenvalue problem, it is a good idea first to reduce a general matrix by means of a similarity transformation to a Hessenberg-type matrix (in the case of a symmetric matrix to a 3-diagonal matrix) and then use the most effective iteration methods. Such a preliminary reduction can be attained by means of a chain of similarity transformations involving elementary matrices, matrices of rotation, or matrices of reflection.

History. The concept of matrix was introduced by W. Hamilton and A. Cayley in the mid-19th century. The foundations of this theory were created by K. Weierstrass and F. Frobenius (second half of the 19th century and early 20th). I. A. Lappo Danilevskii developed the theory of analytic functions of several matrix arguments and applied it to the study of systems of differential equations with analytic coefficients. Matrix notation has become widespread in modern mathematics and its applications. The matrix calculus is developing toward constructing effective algorithms for the numerical solutions of fundamental problems.

REFERENCES

Smirnov V. I. Kurs vysshei matematiki, 9th ed., vol. 3, part 1. Moscow, 1967.
Mal’tsev, A. I. Osnovy lineinoi algebry, 3rd ed. Moscow, 1970.
Gantmakher, F. R. Teoriia matrits, 3rd ed. Moscow, 1967.
Wilkinson, J. H. Algebraicheskaia problema sobstvennykh znacheniL Moscow, 1970. (Translated from English.)
Faddeev, D. K., and V. N. Faddeeva. Vychislitel’nye melody lineinoi algebry, 2nd ed. Moscow-Leningrad, 1963.
Voevodin, V. V. Chislennye metody algebry: Teoriia i algorifmy. Moscow, 1966.
Lappo-Danilevskii, I. A. Primenenie funklsii ol matrits k leorii lineinykh sislem obyknovennykh differential’nykh uravnenii. Moscow, 1957.
Frazer, R. A., W. Duncan, and A. Collar. Teoriia matrits i ee prilozheniia k differential’nym uravneniiam i dinamike. Moscow, 1950. (Translated from English.)
Wasow, W., and G. Forsyth. Raznostnye metody resheniia differentsial’nykh uravnenii v chastnykh proizvodnykh. (Translated from English.) Moscow, 1963.

V. N. FADDEEVA

Matrix

in printing:

(1) An interchangeable element of a casting mold with a recessed (sometimes photographic) image of a letter or symbol used in the casting of characters or lines. A matrix is a metal bar with the face of a letter or sign stamped (by pressure of a punch) or engraved on one of its edges. Characters or lines with a raised printing surface are formed on a matrix pressed to the mold by filling the cavities of the casting mold and face with liquid alloy. A distinction is made between typecasting, Linotype, and Monotype matrices, depending on the type of machine used to cast letters or lines.

A typecasting matrix is a steel bar of rectilinear cross section and a recessed image of a single letter or symbol. A set of typecasting matrices makes it possible to cast on a typecasting machine all letters of a single typeface used for manual composing.

In a Linotype machine a matrix line, which is mounted in front of the slit of a casting mold, is composed from individual matrices stored in the magazine. After the mold is filled with alloy, a solid metal line of type is formed.

In a Monotype machine a set of matrices is collected in a matrix frame. During casting, the required matrix is fitted over the slit of a casting mold. A line of type on a Monotype machine, unlike that on a Linotype machine, is formed from separate letters. The Monotype matrix has an aperture for threading onto the pivot of the matrix frame and a conical depression for precise fixing and pressing of the matrix to the casting mold.

In phototypesetting machines, matrices are used in which recessed images of characters are replaced by photographic images.

(2) The recessed imprint from a raised printing plate on plastic material (cardboard, plastic, and so on), used to produce stereo-type copies of the plate.

G. S. ERSHOV

matrix

[′mā·triks] (analytical chemistry) The analyte as considered in terms of its being an assemblage of constituents, each with its own properties. (computer science) A latticework of input and output leads with logic elements connected at some of their intersections. (electronics) The section of a color television transmitter that transforms the red, green, and blue camera signals into color-difference signals and combines them with the chrominance subcarrier. Also known as color coder; color encoder; encoder. The section of a color television receiver that transforms the color-difference signals into the red, green, and blue signals needed to drive the color picture tube. Also known as color decoder; decoder. (engineering) A recessed mold in which something is formed or cast. (graphic arts) In a type-casting machine, the portion of the mold that forms the letter face. A heavy, unsized, unfinished paper that is used for molds for stereotype plates. A master negative from which characters are projected in a photocomposition process. (histology) The intercellular substance of a tissue. Also known as ground substance. The epithelial tissue from which a toenail or fingernail develops. (materials) A binding agent used to make an agglomerate mass. (mathematics) A rectangular array of numbers or scalars from a vector space. (metallurgy) The principal component of an alloy. The precisely shaped form used as the cathode in electroforming. (mycology) The substrate on or in which fungus grows. (petrology) The continuous, fine-grained material in which large grains of a sediment or sedimentary rock are embedded. Also known as groundmass.

matrix

1. In mortar, the cement paste in which the fine aggregate particles are embedded. 2. In concrete, the mortar in which the coarse aggregate particles are embedded.

matrix

1. Anatomy the thick tissue at the base of a nail from which a fingernail or toenail develops 2. the intercellular substance of bone, cartilage, connective tissue, etc. 3. a. the rock material in which fossils, pebbles, etc., are embedded b. the material in which a mineral is embedded; gangue 4. (formerly) a mould used in the production of gramophone records. It is obtained by electrodeposition onto the master 5. a bed of perforated material placed beneath a workpiece in a press or stamping machine against which the punch operates 6. Metallurgya. the shaped cathode used in electroforming b. the metal constituting the major part of an alloy c. the soft metal in a plain bearing in which the hard particles of surface metal are embedded 7. the main component of a composite material, such as the plastic in a fibre-reinforced plastic 8. Maths a rectangular array of elements set out in rows and columns, used to facilitate the solution of problems, such as the transformation of coordinates. Usually indicated by parentheses: (

abc

def

) 9. Computing a rectangular array of circuit elements usually used to generate one set of signals from another 10. Obsolete the womb

Matrix

[FidoNet] 1. What the Opus BBS software and sysops callFidoNet.

2. Fanciful term for a cyberspace expected to emerge fromcurrent networking experiments (see network, the).

3. The totality of present-day computer networks.

matrix

An array of elements in row and column form. See x-y matrix.

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Bibliography

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REFERENCE

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REFERENCES

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Synonyms for matrix

noun a hollow device for shaping a fluid or plastic substance

Synonyms

Synonyms for matrix

noun (mathematics) a rectangular array of quantities or expressions set out by rows and columns

Related Words

noun (geology) amass of fine-grained rock in which fossils, crystals, or gems are embedded

Related Words

noun an enclosure within which something originates or develops (from the Latin for womb)

Related Words

noun the body substance in which tissue cells are embedded

Synonyms

Related Words

noun the formative tissue at the base of a nail

Related Words

noun mold used in the production of phonograph records, type, or other relief surface

Related Words