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[[Image:lighterstill.jpg]][[Image:Mat9.jpg|right|frame]]
[[Image:lighterstill.jpg]][[Image:Mat9.jpg|right|frame]]
In [[biology]], '''matrix''' (plural: matrices) is the material (or tissue) in [[animal]] or [[plant]] [[cells]], in which more specialized [[structures]] are embedded, and a specific part of the [https://en.wikipedia.org/wiki/Mitochondrion mitochondrion] that is the site of oxidation of organic [[molecules]]. The internal structure of [https://en.wikipedia.org/wiki/Connective_tissue connective tissues] is an [https://en.wikipedia.org/wiki/Extracellular_matrix extracellular matrix]. Finger nails and toenails grow from matrices.
In [[mathematics]], a '''matrix''' (plural matrices, or less commonly matrixes) is a rectangular array of [[number]]s. This way, matrices can record other data that depend on multiple parameters. In particular they are used to keep track of the coefficients of multiple linear equations. Matrices are closely connected to linear transformations, which are higher-dimensional analogs of linear functions, i.e., functions of the form ''f''(''x'') = ''c'' · ''x'', where ''c'' is a constant. This map corresponds to a matrix with one row and column, with entry ''c''. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication, which displays a number of features not encountered in numbers; for example, products of matrices depend on the order of the factors, unlike products of [[real number]]s, say, where [[commutativity|''c - d'' = ''d - c'' for any two numbers ''c'' and ''d''.
In [[mathematics]], a '''matrix''' (plural matrices, or less commonly matrixes) is a rectangular array of [[number]]s. This way, matrices can record other data that depend on multiple parameters. In particular they are used to keep track of the coefficients of multiple linear equations. Matrices are closely connected to linear transformations, which are higher-dimensional analogs of linear functions, i.e., functions of the form ''f''(''x'') = ''c'' · ''x'', where ''c'' is a constant. This map corresponds to a matrix with one row and column, with entry ''c''. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication, which displays a number of features not encountered in numbers; for example, products of matrices depend on the order of the factors, unlike products of [[real number]]s, say, where [[commutativity|''c - d'' = ''d - c'' for any two numbers ''c'' and ''d''.
In the particular case of square matrices, matrices with equal number of columns and rows, more refined data are attached to matrices, notably the [[determinant]], [[inverse matrix|inverse matrices]], which both govern solution properties of the system of linear equation belonging to the matrix, and [[eigenvalues and eigenvectors]].
In the particular case of square matrices, matrices with equal number of columns and rows, more refined data are attached to matrices, notably the [[determinant]], [[inverse matrix|inverse matrices]], which both govern solution properties of the system of linear equation belonging to the matrix, and [[eigenvalues and eigenvectors]].
Matrices are described by the field of matrix theory. The close relationship of matrices with linear transformations makes the former a key notion of linear algebra. Other types of entries, such as elements in more general mathematical fields or even |rings are also used. Matrices consisting of only one column or row are called [[vector]]s, while higher-dimensional, e.g. three-dimensional, arrays of numbers are called [[tensor]]s.
Matrices are described by the field of matrix theory. The close relationship of matrices with linear transformations makes the former a key notion of linear algebra. Other types of entries, such as elements in more general mathematical fields or even |rings are also used. Matrices consisting of only one column or row are called [[vector]]s, while higher-dimensional, e.g. three-dimensional, arrays of numbers are called [[tensor]]s.
== Definition ==
==Definition (Math)==
A ''matrix'' is a rectangular arrangement of [[number]]s. For example,
A ''matrix'' is a rectangular arrangement of [[number]]s. For example,
There are a number of operations that can be applied to modify matrices called ''matrix addition'', ''scalar multiplication'' and ''transposition'' These form the basic techniques to deal with matrices.
There are a number of operations that can be applied to modify matrices called ''matrix addition'', ''scalar multiplication'' and ''transposition'' These form the basic techniques to deal with matrices.
[[Image:Table2bigger.jpg]]
[[Image:Table2bigger.jpg]]
Familiar properties of numbers extend to these operations of matrices: for example, addition is [[commutative]], i.e. the matrix sum does not depend on the order of the summands: '''A''' + '''B''' = '''B''' + '''A'''. The transpose is compatible with addition and scalar multiplication, as expressed by (''c'''''A''')<sup>''T''</sup> = ''c''('''A'''<sup>''T''</sup>) and ('''A''' + '''B''')<sup>''T''</sup> = '''A'''<sup>''T''</sup> + '''B'''<sup>''T''</sup>. Finally, ('''A'''<sup>''T''</sup>)<sup>''T''</sup> = '''A'''.[http://en.wikipedia.org/wiki/Matrix_(mathematics)]
Familiar properties of numbers extend to these operations of matrices: for example, addition is [[commutative]], i.e. the matrix sum does not depend on the order of the summands: '''A''' + '''B''' = '''B''' + '''A'''. The transpose is compatible with addition and scalar multiplication, as expressed by (''c'''''A''')<sup>''T''</sup> = ''c''('''A'''<sup>''T''</sup>) and ('''A''' + '''B''')<sup>''T''</sup> = '''A'''<sup>''T''</sup> + '''B'''<sup>''T''</sup>. Finally, ('''A'''<sup>''T''</sup>)<sup>''T''</sup> = '''A'''.[https://en.wikipedia.org/wiki/Matrix_(mathematics)]
==All Definitions==
*'''I. A supporting or enclosing structure'''.
:1. The womb; the uterus of a mammal. Also (later esp. of an oviparous animal): the ovaries and oviducts, or the ovary alone. Now rare. Perh. Obs.
:2. a. A place or medium in which something is originated, produced, or developed; the environment in which a particular activity or process begins; a point of origin and growth. Now chiefly with reference to abstract things. In early use sometimes with reference to minerals, and overlapping with sense.
::b. The pith of a plant. Obs.
::c. '''Anat. and Zool'''. The generative part of a tissue or organ; spec. the epidermal layer which gives rise to specialized structures such as hairs, feathers, and nails.
::d. '''Bot'''. The substrate on which a fungus or a lichen grows or is attached. Now rare.
:3. a. An embedding or enclosing mass; esp. the rock material in which a metal, fossil, gem, etc., is embedded. Cf. GANGUE n.
::b. '''Biol'''. An amorphous or fibrillar material that surrounds cells; esp. the extracellular substance of connective tissue. Also: the ground substance in which structural elements (e.g. of a shell, cell wall, etc.) are embedded.
::c. A fine material used to bind together the coarser particles of a composite (usually artificial) substance; (Building) lime or fine cement.
::d.'''Cell Biol'''. The ground substance of a cell or organelle; (now) esp. the substance contained within the inner membrane of a mitochondrion. Cf. PROTOPLASM n.
::e. '''Biochem'''. and '''Pharmacol'''. A material that supports or immobilizes a reagent, esp. in separation procedures; a material used to retain a drug for controlled release.
:4. a. The elements which make up a particular system, regarded as an interconnecting network. Freq. with distinguishing word, as political matrix, social matrix, etc.
::b. '''Business'''. An organizational structure in which two or more lines of reporting, responsibility, or communication run through the same individual (often used to supplement a traditional hierarchical structure of organization); spec. such a structure in which project teams are formed of staff drawn from separate departments or functions within the organization. Freq. attrib., esp. in matrix management, matrix organization. Cf. LINE n.2 19d.
:5. '''Science Fiction'''. Also Matrix. With the: = CYBERSPACE n.
*'''II. Technical uses.'''
:6. A mould, die, etc.
::a. '''In Printing''': a metal block in which a character is stamped or engraved so as to form a mould for casting a type; the paper squeeze of a form of type, serving as a mould for a type-metal cast. In Coining: an engraved die used to strike a coin or medal. Also (in extended use): any mould in which something is cast or shaped.
::b. The bed or area hollowed out in a slab in which a monumental brass is fixed.
::c. '''Dentistry'''. A plate of metal or impression material which serves as a temporary wall for a cavity of a tooth during filling.
'''
::d. Sound Recording'''. A copy (positive or negative) of an original disc recording, which is used for making other copies; spec. such a copy used as a stamper.
::e. '''Photogr'''. A dyed print in relief used for transferring colour to a final colour print.
:7. a. '''Math'''. A rectangular array of symbols or mathematical expressions arranged in rows and columns, treated as a single entity, and now usually written within round brackets. Also gen.: any similar tabulated arrangement of items. identity, pay-off, row, singular, unit, unitary matrix, etc.: see the first element.
::b. '''Logic'''. An array of symbols representing truth values, giving the result of all possible assignments of truth values to components of a propositional form or proposition; = truth-table n. at TRUTH n. Compounds 4. Also: that part of a truth table which is an array of the total truth-possibilities (see quot. 1965); a set of basic truth tables for a particular system of logic (see quot. 1973). Freq. attrib.
::c. '''Electronics'''. An array of circuit elements whose interconnections form a rectangular lattice or grid; spec. (a) Computing a single layer of cores in a magnetic core memory (now chiefly hist.); (b) chiefly Television and Broadcasting, a circuit designed to produce outputs that are linear combinations, in different proportions, of a number of inputs. Freq. attrib.
::d. '''Computing'''. A rectangular array of potential image points. Chiefly in dot matrix n. at DOT n.1 Compounds 2. See also matrix printer n. at Compounds 2.
::e. '''Computing'''. With the. The global network of electronic communication.
:8. '''Logic'''. An expression that would become a statement if its variables were replaced by constants (i.e. by names of individuals or classes or statements, as appropriate); = propositional function n. at PROPOSITIONAL adj. Special uses. Also (esp. in predicate calculus): a quantifier-free part of a formula (see quots. 1954 , 1971).
[[Category: Mathematics]]
[[Category: Mathematics]]
[[Category: Computer Science]]
[[Category: Computer Science]]