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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,
[[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'''.[http://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]]