minor

Dictionary: mi·nor (mī'nər) pronunciation

adj.

Lesser or smaller in amount, extent, or size.
Lesser in importance, rank, or stature: a minor politician.
Lesser in seriousness or danger: a minor injury.
Law. Being under legal age; not yet a legal adult.
Chiefly British. Relating to or being the younger or junior of two pupils with the same surname.
Of or relating to a secondary area of academic specialization.
Logic. Dealing with a more restricted category.
Music.
1. Relating to or being a minor scale.
2. Less in distance by a half step than the corresponding major interval.
3. Based on a minor scale: a minor key.

One that is lesser in comparison with others of the same class.
Law. One who has not reached full legal age.
1. A secondary area of specialized academic study, requiring fewer courses or credits than a major.
2. One studying in a secondary area of specialization: She is a physics minor.
Logic.
1. A minor premise.
2. A minor term.
Music. A minor key, scale, or interval.
minors Sports. The minor leagues of a sport, especially baseball.

intr.v., -nored, -nor·ing, -nors.
To pursue academic studies in a minor field: minored in music.

[Middle English, from Latin.]

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5min Related Video: minor

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Business Dictionary: Minor

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Person under the age of Majority specified by law (18 to 21 years, depending on the state). Certain contracts, entered into by a minor, are voidable by the minor. Note that the other party is bound; only the minor may void them. For purposes of the kiddie tax, a minor is anyone under age 14.

Real Estate Dictionary: Minor

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A person under the age of Majority specified by law (18 to 21 years, depending on the state).
Example: Contracts for the sale or use of real estate, entered into by a minor, are voidable by the minor. Note that the other party is bound; only the minor may void them.

Thesaurus: minor

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adjective

Below another in standing or importance: inferior, junior, lesser, low, lower², minor-league, petty, secondary, small, subaltern, subordinate, under. Informal smalltime. See over/under.
Not yet a legal adult: underage². See law, youth/age/maturity.

noun

Law

See

Antonyms: minor

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adj

Definition: insignificant, small
Antonyms: adult, greater, important, large, major, significant

adj, n

Definition: person under legal age of maturity
Antonyms: adult, major, senior

Dental Dictionary: minor

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1. a person of either sex under the age of majority, that is, one who has not attained the age at which full civil rights are granted. adj 2. describing a procedure or treatment that is non- or minimally invasive; describing an illness or condition that is temporary and minimally debilitating.

Music Encyclopedia: Minor

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(1) The name given to a scale whose octave species, in its natural form, is built of the following ascending sequence of intervals: T-S-T-T-S-T-T (T= tone, S= semitone). The note chosen to begin the sequence, the key note, becomes part of the name of the scale, i.e.the scale beginning on A is the scale of A minor. A piece or passage whose melodic basis is a minor scale (say, that on A) and whose harmonic basis is the minor triad on the key note of that scale is said to be ‘in A minor’ (lower-case letters, ‘a minor’ or simply ‘a’ are sometimes used to distinguish minor from major).

In the minor scale, some notes are altered chromatically to increase the harmonic or melodic sense of direction: the ‘harmonic minor’ scale has a raised 7th, in accordance with the major triad on the fifth step (the dominant); the melodic minor scale has a raised 6th and a raised 7th when it is ascending, borrowing the leading-note function of the seventh step from the major scale; when descending it is the same as the natural minor scale. See ex.1.

(2) A minor Interval is one a semitone smaller than a major interval of the same name but contains the same number of diatonic scale steps. A minor Triad is a three-note chord which, reckoned from the lowest note, is built of a minor 3rd and a perfect 5th.

Ex.1

Law Encyclopedia: Minor

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This entry contains information applicable to United States law only.

An infant or person who is under the age of legal competence. A term derived from the civil law, which described a person under a certain age as less than so many years. In most states, a person is no longer a minor after reaching the age of 18 (though state laws might still prohibit certain acts until reaching a greater age; e.g., purchase of liquor). Also, less; of less consideration; lower; a person of inferior condition.

Devil's Dictionary: minor

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A cynical view of the world by Ambrose Bierce

adj.

Less objectionable.

Word Tutor: minor

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IN BRIEF: Less in size or importance. Also: Not yet an adult.

There is no such thing as a minor lapse of integrity. — Tom Peters.

Tutor's tip: Union rules forbade a "minor" (a person under legal age) to do the work of a "miner" (a person who works in a mine).

Wikipedia: Minor (linear algebra)

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This article is about a concept in linear algebra. For the unrelated concept of "minor" in graph theory, see Minor (graph theory).

In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which in turn are useful for computing both the determinant and inverse of square matrices.

1 Detailed definition
2 Nomenclature
3 Cofactors and adjugate or adjoint of a matrix
4 Example
5 Complement
6 Applications
7 Multilinear algebra approach
8 References

Detailed definition

Let A be an m × n matrix and k an integer with 0 < k ≤ m, and k ≤ n. A k × k minor of A is the determinant of a k × k matrix obtained from A by deleting m − k rows and n − k columns.

Since there are

${m \choose k}$ (read "m choose k")

ways to choose k rows from m rows, and there are

${n \choose k}$

ways to choose k columns from n columns, there are a total of

${m \choose k} \cdot {n \choose k}$

minors of size k × k.

Nomenclature

The (i,j) minor (often denoted M_ij) of an n × n square matrix A is defined as the determinant of the (n − 1) × (n − 1) matrix formed by removing from A its i^th row and j^th column. An (i,j) minor is also referred to as (i,j)^th minor, or simply i,j minor.

A minor that is formed by removing only one row and column from a square matrix A (such as M_ij) is called a first minor. When two rows and columns are removed, this is called a second minor.^[1]

Cofactors and adjugate or adjoint of a matrix

The (i,j) cofactor C_ij of a square matrix A is just (−1)^{i + j} times the corresponding (n − 1) × (n − 1) minor M_ij:

C_ij = (−1)^{i + j} M_ij

The cofactor matrix of A, or matrix of A cofactors, typically denoted C, is defined as the n×n matrix whose (i,j) entry is the (i,j) cofactor of A.

The transpose of C is called the adjugate or classical adjoint of A. (In modern terminology, the "adjoint" of a matrix most often refers to the corresponding adjoint operator.) Adjugate matrices are used to compute the inverse of square matrices.

Example

For example, given the matrix

$\begin{pmatrix} \,\,\,1 & 4 & 7 \\ \,\,\,3 & 0 & 5 \\ -1 & 9 & \!11 \\ \end{pmatrix}$

suppose we wish to find the cofactor C₂₃. The minor M₂₃ is the determinant of the above matrix with row 2 and column 3 removed (the following is not standard notation):

$\begin{vmatrix} \,\,1 & 4 & \Box\, \\ \,\Box & \Box & \Box\, \\ -1 & 9 & \Box\, \\ \end{vmatrix}$ yields $\begin{vmatrix} \,\,\,1 & 4\, \\ -1 & 9\, \\ \end{vmatrix} = (9-(-4)) = 13$

where the vertical bars around the matrix indicate that the determinant should be taken. Thus, C₂₃ is (-1)²⁺³ M₂₃ $= -13. \!\$

Complement

The complement, C, of a minor, M, of a square matrix, A, is formed by the determinant of the matrix A from which all the rows and columns associated with M have been removed. The complement of the first minor of an element a_ij is merely that element.^[2]

Applications

The cofactors feature prominently in Laplace's formula for the expansion of determinants. If all the cofactors of a square matrix A are collected to form a new matrix of the same size and then transposed, one obtains the adjugate of A, which is useful in calculating the inverse of small matrices.

Given an m × n matrix with real entries (or entries from any other field) and rank r, then there exists at least one non-zero r × r minor, while all larger minors are zero.

We will use the following notation for minors: if A is an m × n matrix, I is a subset of {1,...,m} with k elements and J is a subset of {1,...,n} with k elements, then we write [A]_I,J for the k × k minor of A that corresponds to the rows with index in I and the columns with index in J.

If I = J, then [A]_I,J is called a principal minor.
If the matrix that corresponds to a principal minor is a quadratic upper-left part of the larger matrix (i.e., it consists of matrix elements in rows and columns from 1 to k), then the principal minor is called a leading principal minor. For an n × n square matrix, there are n leading principal minors.
For Hermitian matrices, the principal minors can be used to test for positive definiteness.

Both the formula for ordinary matrix multiplication and the Cauchy-Binet formula for the determinant of the product of two matrices are special cases of the following general statement about the minors of a product of two matrices. Suppose that A is an m × n matrix, B is an n × p matrix, I is a subset of {1,...,m} with k elements and J is a subset of {1,...,p} with k elements. Then

$[\mathbf{AB}]_{I,J} = \sum_{K} |\mathbf{A}|_{I,K} |\mathbf{B}|_{K,J}\,$

where the sum extends over all subsets K of {1,...,n} with k elements. This formula is a straightforward corollary of the Cauchy-Binet formula.

Multilinear algebra approach

A more systematic, algebraic treatment of the minor concept is given in multilinear algebra, using the wedge product: the k-minors of a matrix are the entries in the kth exterior power map.

If the columns of a matrix are wedged together k at a time, the k × k minors appear as the components of the resulting k-vectors. For example, the 2 × 2 minors of the matrix

$\begin{pmatrix} 1 & 4 \\ 3 & \!\!-1 \\ 2 & 1 \\ \end{pmatrix}$

are −13 (from the first two rows), −7 (from the first and last row), and 5 (from the last two rows). Now consider the wedge product

$(\mathbf{e}_1 + 3\mathbf{e}_2 +2\mathbf{e}_3)\wedge(4\mathbf{e}_1-\mathbf{e}_2+\mathbf{e}_3)$

where the two expressions correspond to the two columns of our matrix. Using the properties of the wedge product, namely that it is bilinear and

$\mathbf{e}_i\wedge \mathbf{e}_i = 0$

and

$\mathbf{e}_i\wedge \mathbf{e}_j = - \mathbf{e}_j\wedge \mathbf{e}_i,$

we can simplify this expression to

$-13 \mathbf{e}_1\wedge \mathbf{e}_2 -7 \mathbf{e}_1\wedge \mathbf{e}_3 +5 \mathbf{e}_2\wedge \mathbf{e}_3$

where the coefficients agree with the minors computed earlier.

References

^ Burnside, William Snow & Panton, Arthur William (1886) Theory of Equations: with an Introduction to the Theory of Binary Algebraic Form.
^ Bertha Jeffreys, Methods of Mathematical Physics, p.135, Cambridge University Press, 1999 ISBN 0521664020.

Topics related to linear algebra

Scalar · Vector · Vector space · Vector projection · Linear span · Linear map · Linear projection · Linear independence · Linear combination · Basis · Column space · Row space · Dual space · Orthogonality · Rank · Minor · Kernel (matrix) · Eigenvalue, eigenvector and eigenspace · Least squares regressions · Outer product · Inner product space · Dot product · Transpose · Gram–Schmidt process · Matrix decomposition

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Translations: Minor

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Dansk (Danish)
adj. - mindre, små-, underordnet
n. - umyndig, mindreårig
v. intr. - tage bifag/sidefag

idioms:

minor axis lille akse
minor key moltoneart
minor league liga med klubber, som ikke er i første række
minor party den mindre/yngre del

Nederlands (Dutch)
van ondergeschikt belang, minderjarige, minderjarig, gering

Français (French)
adj. - léger, (Mus) mineur, (US, Univ) secondaire, (GB, École) junior (arch)
n. - (Jur) mineur, (US, Univ) matière secondaire
v. intr. - (US, Univ) prendre qch en matière secondaire

idioms:

minor axis petit axe
minor in (US, Univ) étudier (qch) comme matière secondaire ou sous-dominante
minor key clé mineure, (Mus) en mineur
minor league (US, Sport) division secondaire
minor party parti minoritaire

Deutsch (German)
n. - Minderjähriger, Nebenfach
adj. - weniger wichtig, kleiner, Neben..., geringfügig, nebensächlich, minderjährig
v. - etw. als Nebenfach haben, im Nebenfach studieren

idioms:

minor axis (Geom.) kleine Achse
minor in etwas als Nebenfach haben
minor key Molltonart
minor league untere Liga
minor party kleine Partei, Minderheitspartei

Ελληνική (Greek)
n. - ανήλικος, (μουσ.) ελάσσων, μινόρε
adj. - μικρότερος, δευτερεύων, επουσιώδης, μικρός

idioms:

minor axis ελάχιστος ελλειψοειδής άξονας
minor key (μουσ.) ελάσσων τόνος, μινόρε
minor league (ΗΠΑ) (στο φούτμπολ, μπέιζμπολ, κ.λπ.) δεύτερη κατηγορία
minor party μικρό κόμμα

Italiano (Italian)
minorenne, materia secondaria, secondario, minore

idioms:

minor axis asse minore
minor key chiave minore
minor league lega minore
minor party partito minuscolo

Português (Portuguese)
n. - menor (f) (de idade), tom menor (m) (Mús.)
adj. - menor, secundário

idioms:

minor axis eixo perpendicular ao eixo principal (Geom.)
minor key terça menor (f) (Mús.)
minor league time de segunda divisão (m)
minor party partido minoritário

Русский (Russian)
второстепенный, несовершеннолетний, минорный, малый, младший из двух братьев, изучать что-л. в качестве второй специальности

idioms:

minor axis малая ось
minor key минорная нота
minor league второстепенная спортивная лига
minor party небольшая политическая партия

Español (Spanish)
adj. - menor, secundario, de segundo orden, más pequeño
n. - menor, asignatura secundaria, materia secundaria
v. intr. - elegir como asignatura o materia secundaria

idioms:

minor axis eje menor
minor in estudiar una materia subsidiaria
minor key tono menor (mús.)
minor league liga secundaria de clubes profesionales, liga menor
minor party partido minoritario

Svenska (Swedish)
n. - minderårig, tillvalsämne, student som har ngt. som tillvalsämne, moll, undersats
adj. - mindre, små-, minderårig

中文（简体）(Chinese (Simplified))
较小的, 未成年的, 二流的, 未成年人, 副修科目, 小调, 小音阶, 辅修

idioms:

minor axis 短轴
minor key 小调
minor league 小职业球队联盟
minor party 少数党, 小政党

中文（繁體）(Chinese (Traditional))
adj. - 較小的, 未成年的, 二流的
n. - 未成年人, 副修科目, 小調, 小音階
v. intr. - 輔修

idioms:

minor axis 短軸
minor key 小調
minor league 小職業球隊聯盟
minor party 少數黨, 小政黨

한국어 (Korean)
adj. - 중요하지 않은, 작은, 미성년의, 아래 사람의, (음악의) 단조의
n. - 성년이 아직 안된 사람, 부전공 과목, 소전제
v. intr. - 부전공을 하다

日本語 (Japanese)
adj. - 小さいほうの, 重要でない, 短調の, 軽症の, 年少の, 未成年の, 短音程の
n. - 未成年者, 副専攻科目, 短調

idioms:

minor axis 短軸
minor key 短調, 陰気な気分
minor league マイナーリーグ
minor party 少数党

العربيه (Arabic)
‏(الاسم) قاصر (صفه) أصغر, أقل, صغير‏

עברית (Hebrew)
adj. - ‮משני, זוטר, קטן, צעיר, טפל, לא רציני, מינורי, זעיר‬
n. - ‮קטין, מקצוע לימודים משני של סטודנט (ארה"ב)‬
v. intr. - ‮למד (נושא) כמקצוע לימודים משני (סטודנט)‬

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minor

Contents

Detailed definition

Nomenclature

Cofactors and adjugate or adjoint of a matrix

Example

Complement

Applications

Multilinear algebra approach

References