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meaning 'a mould in which something is shaped' (and other technical meanings), is pronounced may-triks and has the plural form matrices (may-tri-seez) or matrixes.

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An r×c matrix consists of a rectangular array with r rows and c columns, in which the elements are either numbers or algebraic expressions. Example matrices (the plural form) are:

. When the array is not written out in full, a matrix is usually denoted by a bold-face capital letter, e.g. X, or by a typical element (or entry) from the array, shown in curly brackets, e.g. {x_jk}, where x_jk is the element in the jth row and kth column of the matrix. If r=c the matrix is square.

If a matrix X={x_jk} is multiplied by the real number s, then the result is the matrix sX, in which the element in the jth row and kth column is sx_jk. In this context a real number s is often referred to as a scalar.

Two matrices, A and B, can be multiplied together only if the number of columns of one matrix is equal to the number of rows of the other matrix. If A is an m×n matrix and B is an n×p matrix then the product AB is an m×p matrix. However, if p≠m then the product BA does not exist. The rule for the construction of the product is as follows. Let e_jk denote the element in the jth row and kth column of the product AB, with a_jk and b_jk denoting typical elements in A and B. Then e_jk is given by


.  If A and B have the same values of r and c and if a_jk=b_jk for all j and k, then A=B.

A diagonal matrix is a square matrix with all elements equal to 0, except for those on the leading diagonal (which runs from top-left to bottom-right). This diagonal is also called the main diagonal. A matrix (not necessarily square) in which all the entries are equal on every negatively sloping diagonal is a Toeplitz matrix. For example:


. An identity matrix, usually denoted by I, is a diagonal matrix with on-diagonal elements all equal to 1. The size of an identity matrix may be indicated using a suffix: I_n is an n×n identity matrix.

The transpose of an m×n matrix M is the n×m matrix formed by interchanging the elements of the rows and columns of M. It is denoted by M′. The jth row of M′ is the transpose of the jth column of M and vice versa


. If a square matrix S, with typical element s_jk, is equal to its transpose, S′, then it is a symmetric matrix satisfying s_jk=s_kj, for all j, k.A square matrix that is not symmetric is an asymmetric matrix. If a square matrix S satisfies the equation SS=S then it is idempotent. The product SS may be written as S². If it exists, the inverse of a square matrix, S, is denoted by S⁻¹. It satisfies the relations that SS⁻¹=S⁻¹S=I.Only square matrices can have an inverse (but see 'generalized inverse' below). If S⁻¹ exists then it will be the same size as S. A matrix that has an inverse is said to be non-singular (or regular, or invertible). A square matrix without an inverse is said to be singular.

A square matrix is described as being an upper triangular matrix if all the elements below the leading diagonal are zero, or as a lower triangular matrix if all the elements above the leading diagonal are zero. The matrices U and L are examples:


. A generalized inverse (also called a Moore–Penrose inverse) of the m×n matrix M is any n×m matrix M⁻ satisfying MM⁻M=M.If a matrix M is multiplied by its transpose (to give either MM′ or M′M) then the result is a symmetric matrix.

If M is square and the product MM′ is an identity matrix, then M′=M⁻¹ and M is said to be an orthogonal matrix.

A matrix with just one row is called a row vector. A matrix with just one column is called a column vector. Column vectors are usually denoted with a bold-face lower-case letter, e.g. x; row vectors are written as their transpose, e.g. x′. A vector with a single element (i.e. a 1×1 matrix) is a scalar.

Vectors multiply together in the same way as matrices (see above). Thus, if v is an n×1 column vector, and v′ is its transpose, then the product vv′ is an n×n symmetric matrix, and the product v′v is a scalar.

The set of n×1 vectors v₁, v₂,..., v_m is linearly independent if the only values of the scalars a₁, a₂,..., a_m for which


, where 0 is an n×1 vector with every element equal to 0, is a₁=a₂=...=a_m=0. If the set is not linearly independent then it is linearly dependent, in which case there are values for the scalars a₁, a₂,..., a_m, not all equal to 0, such that


A linearly independent set with two or more vectors satisfies the requirement that at least one of the vectors, v_k, say, is a linear combination of the others, i.e.


, for some scalars b₁, b₂,..., b_k−1, b_k+1,..., b_m.

The rank of a matrix is the maximum number of linearly independent rows, which is the same as the maximum number of linearly independent columns. Thus the rank of a matrix is equal to that of its transpose. If a matrix has r rows and c columns, with r≤c, then the rank is≤r; if r>c then the rank is≤c. If the rank is equal to the smaller of r and c then the matrix is of full rank.

If A is a square matrix, x is a column vector not equal to 0, and λ is a scalar such that Ax=λx,then x is an eigenvector of A and λ is the corresponding eigenvalue. Eigenvectors and eigenvalues are also referred to as characteristic vectors and characteristic values. If x is the column vector (x₁ x₂...x_n)′ and A is an n×n symmetric matrix with typical element a_jk, then the product x′Ax, which is a scalar, is described as a quadratic form because it is equal to


, which is a linear combination of all the squared terms (such as x₁²) and cross-products (such as x₁x₂).

A symmetric matrix A is a positive definite matrix if x′Ax>0 for all non-zero x; it is a positive semi-definite matrix if x′Ax≥0 for all x and there is at least one non-zero x for which x′Ax=0.

The trace of a square matrix is the sum of the terms on the leading diagonal.

The determinant of a 2×2 square matrix, A, is written as |A| or det(A), and is given by |A|=a₁₁a₂₂−a₁₂a₂₁.The determinant of a larger matrix is defined recursively in terms of cofactors. The cofactor of the entry a_jk is equal to the product of (−1)^j+k and the determinant A_jk of the matrix obtained by eliminating the jth row and kth column of A. The recursive definition is


. In fact


if k=l (otherwise the sum is 0). Similarly,


if j=l and is otherwise 0. Thus, for a 3×3 matrix, A,|A|=a₁₁(a₂₂a₃₃−a₂₃a₃₂)−a₁₂(a₂₁a₃₃−a₂₃a₃₁)+a₁₃(a₂₁a₃₂−a₂₂a₃₁).The eigenvalues of a square matrix A are the roots of the characteristic equationdet(A−λI)=0.

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For more information on matrix, visit Britannica.com.

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

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1. Paper mold of a printing plate made from a type form; also called mat. A matrix is used primarily for newspaper reproductions where the advertiser needs an inexpensive duplicate and wants to preserve the original for future use. See also stereotype.

2. Small brass mold of a type character, used in the machine casting of type.

Mathematical term describing a rectangular array of elements (numerical data, parameters or variables). Each element within a matrix has a unique position, defined by the row and column.

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noun

A hollow device for shaping a fluid or plastic substance: cast, form, mold. See surface/depth.

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.

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[Ge]

1. The material or sediment in which cultural debris is contained; the surrounding deposit in which archaeological finds are situated.

2. Harris matrix.

3. The main metal component of an alloy.

1. A substance, situation, or environment which encloses something or from which something originates.

2. The extracellular substance secreted by cells that determines the specialized function of each type of connective tissue.

3. The rectangular array of elements presented in rows and columns, used to facilitate the solution of problems.

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(pl. matrices or matrixes)

1. a medium or place wherein something is formed, develops, or is embedded or enclosed.
2. the medium that forms, or supports, the stationary phase in certain separative procedures, as in chromatography or zone electrophoresis; the material that forms, or is used to form, the substratum of an immobilized reagent in affinity chromatography.
3. (in animal histology) see extracellular matrix.
4. (in plant histology) the meshwork of soluble substances, largely polysaccharide in nature, occurring in cell walls and in which insoluble microfibrils of cellulose are embedded.
5. see mitochondrial matrix.
6. cytoplasmic matrix; see groundplasm.
7. the substratum, living or nonliving, in or on which a fungus or lichen grows.
8. (in mathematics) any rectangular array of numbers or symbols manipulable algebraically as a single entity and used according to specific rules to facilitate solving of certain kinds of problems, e.g. a set of simultaneous linear equations.

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Pl. matrices [L.]
1. the intercellular substance of a tissue, such as bone matrix.
2. the tissue from which a structure develops, such as hair or nail matrix.
3. a rectangular arrangement of quantities or symbols.

• bone m. (1) — see bone matrix.
• cartilage m. (1) — the intercellular substance of cartilage, consisting of cells and extracellular fibers embedded in an amorphous ground substance.
• claw m. (2) — the claw bed. Called also matrix unguis.
• correlation m. (3) — a square table giving a correlation between each pair of a set of variables. The diagonal elements give the correlation of a variable with itself, namely 1.
• covariance m. — similar to the correlation matrix but gives the variances and covariances.
• m. Gla protein — part of the organic phase of bones; found tightly bound to the bone morphogenetic protein of Urist.
• transition m. (3) — a table of values used in a markov chain mathematical model, giving the probability of a transition from one state to another in a specified time interval.
• m. unguis — see claw matrix (above).

(mā'triks)
n

1. an intergranular substance that acts somewhat as a cementing material for other particles; for example, zinc phosphate cement is made of undissolved zinc oxide particles, surrounded and held or cemented together by phosphate compounds. The phosphate compounds make up the matrix n 2. a mechanical or artificial wall that completes the mold into which plastic material is inserted. n 3. a mold into which something is formed. See also bone; splint.

For a list of words related to matrix, see:

• Tissue, Fiber, and Integumentary System - matrix: solid material made by cells and secreted around them
• Cell Structure and Function - matrix: connective tissue ground substance
• Mining - matrix: gangue
• Quantities, Relationships, and Operations - matrix: ordered set of elements arranged in rows and columns, esp. one that can be combined with similar arrays having comparable number of rows and columns
• Sculpture Tools and Techniques - matrix: base material to be carved
• Sound Reproduction Technology - matrix: master
• Printing and Typefaces - matrix: mold for casting typefaces
• Order, Hierarchy, and Systems - matrix: arrangement of elements into rows and columns
• Enlightenment, Truth, and the Unknowable - matrix: point of origin

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See words rhyming with "matrices."

  See crossword solutions for the clue Matrix.

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Dansk (Danish)
n. - matrice, støbeform

Nederlands (Dutch)
matrijs, baarmoeder, iets waarin iets anders zich ontwikkelt, vorm, knipinstrument, stempel, inbeddingsmateriaal, elektrisch circuit, hoofdzin die bijzin bevat, matrix

Français (French)
n. - (Anat, Comput, Ling, Math, Imprim, Tech) matrice, (Minér) gangue

Deutsch (German)
n. - Matrize, Matrix

Ελληνική (Greek)
n. - μήτρα (κν. καλούπι), (μαθημ.) μήτρα, μητρώο, (ανατ.) μεσοκυττάρια ουσία

Italiano (Italian)
matrice

Português (Portuguese)
n. - matriz (f), útero (m), madre (f)

Русский (Russian)
матрица, матка

Español (Spanish)
n. - molde, matriz

Svenska (Swedish)
n. - matris, gjutform, matrix (anat.), källa, malmåder

中文（简体）(Chinese (Simplified))
母体, 子宫, 基础, 基质

中文（繁體）(Chinese (Traditional))
n. - 母體, 子宮, 基礎, 基質

한국어 (Korean)
n. - 지형, 자궁, 형태

日本語 (Japanese)
n. - 母体, 基盤, 細胞間質, 爪母基, マトリックス, 母型文, 母型, 原盤
v. - マトリックス化する

العربيه (Arabic)
‏(الاسم) رحم الأم, قالب‏

עברית (Hebrew)
n. - ‮אימה, מטריצה, טבלה‬

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