Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definition: Determinant |
DeterminantAdjective1. Having the power or quality of deciding; "the crucial experiment"; "cast the deciding vote"; "the determinative (or determinant) battle". Noun1. A determining or causal element or factor; "education is an important determinant of one's outlook on life". 2. A square matrix used to solve simultaneous equations. Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Date "determinant" was first used in popular English literature: sometime before 1790. (references) |
| Domain | Definition |
Computing | A square array of elements that represents the sum of certain products of these elements, used to solve simultaneous equations, in vector studies, etc. Source: European Union. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
In linear algebra, the determinant is a function that associates a scalar to every square matrix. For instance, the 2-by-2 matrix
has determinant
A formula for bigger matrices will be given below.
- .
The determinant of A is also sometimes denoted by |A|, but this notation should be avoided as it is also used to denote other matrix functions, such as the square root of AA*.
History and applications
Historically, determinants were considered before matrices. Originally, a determinant was defined as a property of a system of linear equations. The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero). In this sense, two-by-two determinants were considered by Cardano at the end of the 16th century and larger ones by Leibniz about 100 years later.
Determinants are used to characterize invertible matrices, and to explicitly describe the solution to a system of linear equations with Cramer's rule. It can be used to find the eigenvalues of the matrix A through the characteristic polynomial p(x) = det(A-xIn).
One often thinks of the determinant as assigning a number to every sequence of n vectors in Rn, by using the square matrix whose columns are the given vectors. With this understanding, the sign of the determinant of a basis can be used to define the notion of orientation in Euclidean spaces.
Determinants are used to calculate volumes in vector calculus: the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if the linear map f : Rn -> Rn is represented by the matrix A, and S is any measurable subset of Rn, then the volume of f(S) is given by |det(A)| × volume(S). More generally, if the linear map f : Rn -> Rm is represented by the m-by-n matrix A, and S is any measurable subset of Rn, then the n-dimensional volume of f(S) is given by √(det(ATA)) × volume(S).
Definition and Computation
Suppose A = (Ai,j) is a square matrix.
If A is a 1-by-1 matrix, then det(A) = A1,1. If A is a 2-by-2 matrix, then det(A) = A1,1 · A2,2 - A2,1 · A1,2. For a 3-by-3 matrix A, the formula is more complicated:
For a general n-by-n matrix, the determinant was defined by Gottfried Leibniz with what is now known as the Leibniz formula:
- det(A) = A1,1·A2,2·A3,3 + A1,3·A3,2·A2,1 + A1,2·A2,3·A3,1
- - A3,1·A2,2·A1,3 - A1,1·A2,3·A3,2 - A1,2·A2,1·A3,3
The sum is computed over all permutations σ of the numbers {1,...,n} and sgn(σ) denotes the signature of the permutation σ: +1 if σ is an even permutation and -1 if it is odd. See symmetric group for an explanation of even/odd permutations.
This formula contains n summands and is therefore impractical to use if n is bigger than 3.
In general, determinants can be computed with the Gauss algorithm using the following rules:
Explicitly, starting out with some matrix, use the last three rules to convert it into a triangular matrix, then use the first rule to compute its determinant.
- If A is a triangular matrix, i.e. Ai,j = 0 whenver i > j, then det(A) = A1,1·A2,2·...·An,n
- If B results from A by exchanging two rows or columns, then det(B) = - det(A)
- If B results from A by multiplying one row or column with the number c, then det(B) = c · det(A)
- If B results from A by adding a multiple of one row or column to another row or column, then det(B) = det(A).
It is also possible to expand a determinant along a row or column using Laplace's formula, which is efficient for relatively small matrices. To do this along row i, say, we write
where the Ci,j represent the matrix cofactors, i.e. Ci,j is (-1)i+j times the determinant of the matrix that results from A by removing the i-th row and the j-th column.
Properties
The determinant is a multiplicative map in the sense that
This is generalized by the Cauchy-Binet formula to products of non-square matrices.
- det(AB) = det(A)det(B) for all n-by-n matrices A and B.
It is easy to see that det(rIn)=rn and thus
If A is invertible, then
- det(rA) = rn det(A) for all n-by-n matrices A and all scalars r.
A matrix and its transpose have the same determinant:
- det(A-1)=det(A)-1.
If A and B are similar, i.e. if there exists an invertible matrix X such that A = X-1BX, then by the multiplicative property,
- det(A) = det(AT).
Because of this, one may define the determinant of a linear map f : V -> V (where V is a finite-dimensional vector space) by choosing a basis for V, describing f as a matrix relative to this basis, and taking the determinant of this square matrix. The result will not depend on the basis chosen.
- det(A) = det(B).
There exist matrices which have the same determinant but are not similar.
If A is a square n-by-n matrix with real or complex entries and if λ1,...,λn are the (complex) eigenvalues of A listed according to their algebraic multiplicities, then
This follows from the fact that A is always similar to its Jordan normal form, an upper triangular matrix with the eigenvalues on the main diagonal.
- det(A) = λ1·λ2·...·λn.
From the connection between the determinant and the eigenvalues, one can derive a connection between the trace function, the exponential function, and the determinant:
- det(exp(A)) = exp(tr(A)).
Derivative
The determinant of real square matrices is a polynomial function from Rn×n to R, and as such is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:
where adj(A) denotes the adjugate of A. In particular, if A is invertible, we have
- d det(A) = tr(adj(A) dA)
or, more colloquially,
- d det(A) = det(A) tr(A-1 dA)
if the entries in the matrix X are sufficiently small. The special case where A is equal to the identity matrix I yields
- det(A + X) - det(A) ≈ det(A) tr(A-1 X)
- det(I + X) ≈ 1 + tr(X).
Generalizations
It makes sense to define the determinant for matrices whose entries come from any commutative ring. The computation rules, the Leibniz formula and the compatibility with matrix multiplication remain valid, except that now a matrix A is invertible if and only if det(A) is an invertible element of the ground ring.
Abstractly, one may define the determinant as a certain anti-symmetric multilinear map as follows: if R is a commutative ring and M = Rn denotes the free R-module with n generators, then
is the unique map with the following properties:
- det : Mn -> R
Linear Algebraists prefer to use the multilinear map approach to define determinant, whereas algebraists prefer to use the Leibniz formula.
- det is R-linear in each of the n arguments.
- det is anti-symmetric, meaning that if two of the n arguments are equal, then the determinant is zero.
- det(e1,..,en) = 1, where ei is that element of M which has a 1 in the i-th coordinate and zeros elsewhere.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Determinant."
Synonyms: DeterminantSynonyms: crucial (adj), deciding(a) (adj), determining(a) (adj), causal factor (n), determinative (n), determiner (n), determining factor (n). (additional references) |
Crosswords: Determinant |
| English words defined with "determinant": causal factor, crucial ♦ deciding, determinative, determiner, determining, determining factor ♦ harmonic ♦ nonsingular matrix ♦ organicism ♦ singular matrix, Skew symmetrical determinant. (references) |
| Specialty definitions using "determinant": Antibody Affinity, antigenic determinant ♦ Binding Sites, Antibody ♦ Cognitive Therapy ♦ Disease Progression ♦ eigenvalues, epitope ♦ Gonadal Dysgenesis, 46,XY ♦ idiotype, Immunodominant Epitopes, inverse matrix, inverse of a matrix ♦ latent roots ♦ Nitrohydroxyiodophenylacetate ♦ Plasmids ♦ Resting metabolic rate, RMR. (references) |
| Non-English Usage: "Determinant" is also a word in the following languages with English translations in parentheses. Romanian (decisive, determinant, determinative, determiner), Swedish (determinant), Turkish (determinant). |
| Domain | Title |
Books |
|
Source: compiled by the editor from various references; see credits. | |
| Subject | Topic | Quote |
Health | Significant disease may be a determinant of morbidity. (references) | |
CA 125 is an antigenic determinant detected by radioimmumoassay. (references) | ||
The most important determinant of risk is the destination of the traveler. (references) | ||
Business | The cycle of government financing programs for the procurement of agricultural machinery has been a major determinant of market demand. (references) | |
Especially with the present economic recession price will be the top determinant factor in the sales of the equipment after enforcement of regulations. (references) | ||
Enforcement of standards and regulations by the relevant government agencies would ultimately be the main determinant of air pollution equipment demand. (references) | ||
Economic History | Taiwan | Product brand is an important determinant of price policy. (references) |
Canada | Value for dollar is the predominant purchasing determinant in both the consumer and industrial markets. (references) | |
Panama | This buys high quality trend-setting goods where price is a secondary determinant in the purchasing decision. (references) | |
Trade | Japan | Adherence to JIS is also an important determinant for companies competing on bids in the Japanese government procurement process. (references) |
Source: compiled by the editor from ICON Group International, Inc.; see credits. | ||
| "Determinant" is generally used as a noun (singular) -- approximately 92.95% of the time. "Determinant" is used about 241 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Noun (singular) | 92.95% | 224 | 20,130 |
| Adjective (general or positive) | 7.05% | 17 | 85,106 |
| Total | 100.00% | 241 | N/A |
Source: compiled by the editor from several corpora; see credits.
Expressions using "determinant": antigenic determinant ♦ proximate determinant of fertility ♦ skew symmetrical determinant. Additional references. | |
| Hypenated Usage | |
Ending with "determinant": self-determinant. | |
| Source: compiled by the editor from various references; see credits. | |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Language | Translations for "determinant"; alternative meanings/domain in parentheses. | |
Albanian | vendimtar (conclusive, critical, crucial, decisive, determinative, winning), përcaktues (adherent, designing, determinative), përcaktor. (various references) | |
Arabic | المقرر. (various references) | |
Bulgarian | решаващ (conclusive, crucial, decisive, determinative, last), определящ (determinative, qualificatory, qualifying). (various references) | |
Chinese | 定列式. (various references) | |
Danish | determinant. (various references) | |
Dutch | determinant. (various references) | |
Farsi | تعیین کننده , تصمیم گیرنده , عاجز (Cripple, Incapable, Septum, Unable), جازم . (various references) | |
Finnish | determinantti, epitooppi (antigenic determinant, epitope), antigeeninen determinantti. (various references) | |
French | déterminant (determining). (various references) | |
German | bestimmend (appointing, destining), Determinante. (various references) | |
Greek | Ορίζουσα, καθοριστικόσ (decisive, determinative, determined), συντελεστής προσδιορισμού, προσδιοριστικόσ (attributive), παράγοντασ που αποφασίζει, ορίζουσα. (various references) | |
Hebrew | קוצב (pacemaker, pacesetter). (various references) | |
Hungarian | determináns, döntő (determinate, determinative). (various references) | |
Indonesian | determinasi, determinan, faktor penentu. (various references) | |
Italian | determinativo (definite, destining). (various references) | |
Japanese Kanji | 行列式 . (various references) | |
Japanese Katakana | ぎょうれつしき. (various references) | |
Korean | 결정 (Conclusive, Crucial, Crystal, Crystallizing, Deciding, Decision, Definitive, Determining, irrevocable). (various references) | |
Pig Latin | eterminantday.(various references) | |
Portuguese | determinante (determinative). (various references) | |
Romanian | determinant (decisive, determinative, determiner), decisiv (conclusive, critical, crucial, decisive, definitive, determinative, fateful, final, ultimate, winning), factor determinant (determinative). (various references) | |
Russian | решающий (decisive), определяющий (definative), детерминант определяющий, детерминант. (various references) | |
Serbo-Croatian | koji određuje (qualificatory). (various references) | |
Spanish | determinante (determiner). (various references) | |
Swedish | determinant. (various references) | |
Turkish | determinant, sıfat (adjectival, adjective, attribute, attribution, attributive, capacity, character, epithet, title), hakim olan (possessor, ruling), hükmeden (commanding, dominant, domineering, governing, masterful, regnant, suzerain), etkin olan (in the ascendant, in the ascendent), etken (active, agent, dominant, effective, factor, ingredient), belirleyici faktör. (various references) | |
Ukrainian | вирішальний фактор (determinative), вирішальний (all in all, conclusive, critical, crucial, decisive, determinative, fateful, final, winning), визначальний (denotative, determinative, instrumental, nominating, qualificative, qualificatory), детермінант. (various references) | |
Vietnamese | xác định (determinative), định rõ; định lượng quyết định. (various references) | |
| Source: compiled by the editor from various translation references. | ||
Derivations | |
Words beginning with "determinant": determinantal, determinants. (additional references) | |
| |
"Determinant" is suggested in spellcheckers for the following: determinans, determinantal, determinen, determineth. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| # of Phoneme Matches | Pronunciation | Word(s) rhyming with "determinant" (pronounced duter"mununt) |
| 7 | -er" m u n u n t | permanent, semipermanent. |
| 6 | -m u n u n t | contaminant, immanent, imminent, dominant, eminent, predominant, preeminent, prominent, ruminant. |
| 5 | -u n u n t | abstinent, consonant, continent, covenant, impertinent, dissonant, incontinent, pertinent, recombinant, resonant, subcontinent. |
| 4 | -n u n t | complainant, component, exponent, indignant, lieutenant, malignant, opponent, pennant, pregnant, proponent, remnant, repugnant, stagnant, tenant. |
| 3 | -u n t | agent, aggrandizement, agreement, ailment, alignment, allotment, amazement, ambient, ambivalent, amendment, amusement, ancient, announcement, annulment, antecedent, antidepressant, antigovernment, antioxidant, apartment, apparent, appeasement, applicant, appointment, apportionment, ardent, argent, argument, armament, arraignment, arrangement, arrant, arrogant, absent, absorbent, abundant, accelerant, accident, accompaniment, accomplishment, abandonment, abatement, aberrant, abhorrent, abortifacient, accountant, accouterment, achievement, acknowledgement, acknowledgment, adamant, adherent, adjacent, adjournment, adjustment, adjutant, adolescent, adornment, advancement, advertisement, advisement, afferent, affiant, affluent, ascendant, aspirant, assailant, assessment, assignment, assistant, assortment, astonishment, astringent, atonement, attachment, attainment, attendant, banishment, basement, battlement, belligerent, bemusement, beneficent, benevolent, bereavement, betterment, bewilderment, blandishment, blatant, bombardment, brilliant, buoyant, celebrant, claimant, clairvoyant, Clement, client, coefficient, cogent, cognizant, coherent, coincident, combatant, commandment, commencement, commitment, compartment, competent, complacent, complaisant, complement, compliant, comportment, concealment, concomitant, concurrent, condiment, confident, confinement, confluent, consequent, consignment, consistent, constant, constituent, consultant, containment, contentment, contestant, contingent, convalescent, convenient, convent, convergent, conversant, coolant, copayment, corespondent, cormorant, correspondent, Courant, Crescent, crosscurrent, current, curtailment, ignorant, immigrant, impairment, impatient, impeachment, impediment, implement, important, impotent, impoundment, impoverishment, impressment, imprisonment, improvement, imprudent, inadvertent, incandescent, debarment, debasement, decadent, decedent, decent, declarant, decongestant, defendant, deferment, defiant, deficient, defoliant, delinquent, deodorant, department, dependent, deployment, deportment, depressant, derailment, descendant, descendent, despondent, detachment, detergent, determent, deterrent, detriment, development, deviant, different, diligent, diminishment, disagreement, disappointment, disarmament, disbarment, disbursement, discernment, discordant, discouragement, disenchantment, disenfranchisement, disengagement, disestablishment, disgruntlement, disillusionment, disinfectant, disinvestment, dismantlement, dismemberment, disobedient, dispersant, displacement, dissident, distant, divalent, divergent, divestment, docent, dormant, easement, ebullient, efferent, effervescent, efficient, effluent, elegant, element, elephant, eloquent, embankment, embarrassment, embayment, embellishment, embezzlement, embodiment, emergent, emigrant, emplacement, employment, empowerment, enactment, encampment, enchantment, encirclement, encouragement, encroachment, endangerment, endearment, endorsement, endowment, enforcement, engagement, enhancement, enjoyment, enlargement, enlightenment, enlistment, enrichment, enrollment, enslavement, entanglement, entertainment, enticement, entitlement, entombment, entrant, entrapment, entrenchment, environment, equipment, equivalent, errant, escapement, escarpment, esculent, establishment, estrangement, evanescent, evident, excellent, excitement, excrement, exigent, existent, exorbitant, expectant, expectorant, expedient, experiment, extant, extinguishment, extravagant, exuberant, exultant, Fabricant, fervent, figment, filament, flagrant, flamboyant, flatulent, flippant, fluent, fluorescent, formant, fragment, fragrant, fraudulent, frequent, fulfillment, gallant, garment, garnishment, giant, government, grandiloquent, grandparent, harassment, hesitant, hydrant, incessant, incident, incipient, incitement, inclement, incoherent, incompetent, inconsistent, inconvenient, increment, incumbent, indecent, independent, indictment, indifferent, indigent, indolent, inducement, indulgent, inefficient, infant, informant, infotainment, infrequent, infringement, ingredient, inhabitant, inhalant, inherent, innocent, inpatient, insignificant, insistent, insolent, insolvent, installment, instant, instrument, insufficient, insurgent, integument, intelligent, intercurrent, interdependent, intermittent, internment, intersegment, intolerant, intransigent, invariant, investment, involvement, iridescent, irrelevant, irreverent, irritant, itinerant, jubilant, judgement, judgment, latent, leant, lenient, ligament, litigant, lubricant, lucent, luminescent, luxuriant, magnificent, malevolent, maltreatment, management, measurement, merchant, micromanagement, migrant, militant, miscreant, misgovernment, misjudgment, mismanagement, misstatement, mistreatment, moment, monovalent, monument, mordant, movement, mutant, nascent, negligent, noncombatant, nonexistent, nongovernment, nonmanagement, nonpayment, nonresident, nonviolent, nourishment, nutrient, obedient, observant, obsolescent, occupant, odorant, ointment, omnipotent, omnipresent, omniscient, operant, opulent, ornament, orpiment, outpatient, outplacement, overconfident, overpayment, overstatement, oxidant, pageant, parchment, parent, parliament, participant, patent, patient, pavement, payment, peasant, penchant, pendant, penitent, percipient, persistent, petulant, pheasant, pigment, piquant, placement, pleasant, pliant, poignant, pollutant, postponement, postretirement, potent, preadolescent, precedent, predicament, prejudgment, prepayment, prescient, present, presentment, prevalent, procurement, proficient, pronouncement, propellant, protestant, provident, prudent, prurient, pungent, punishment, pursuant, puzzlement, quadrant, quiescent, quotient, radiant, rampant, readjustment, reagent, realignment, reappointment, reapportionment, rearmament, rearrangement, reassessment, reassignment, recalcitrant, recent, recipient, recruitment, recurrent, redeployment, redevelopment, redundant, reemployment, reenactment, refinement, refreshment, refrigerant, refurbishment, Regent, regiment, registrant, reimbursement, reinforcement, reinstatement, reinvestment, relevant, reliant, reluctant, reminiscent, repayment, repellent, repentant, replacement, replenishment, requirement, resentment, resettlement, resident, resilient, resistant, resplendent, respondent, restatement, resultant, resurgent, retardant, reticent, retirement, retrenchment, reverent, rodent, rudiment, sacrament, salient, seafront, sealant, sediment, segment, sentiment, Sequent, sergeant, serpent, servant, settlement, shipment, significant, silent, solvent, somnolent, statement, stimulant, strident, stringent, student, subsequent, subservient, succulent, sufficient, supergiant, superintendent, supplement, supplicant, suppressant, surfactant, talent, tangent, temperament, tenement, testament, tetravalent, tolerant, torrent, tournament, transcendent, transient, translucent, transparent, treatment, trenchant, trident, triumphant, truant, truculent, tumescent, turbulent, tyrant, undercurrent, underdevelopment, underemployment, undergarment, underpayment, understatement, unemployment, unimportant, unpleasant, unrepentant, urgent, vacant, vagrant, valiant, variant, vehement, verdant, vibrant, vigilant, violent, virulent, warrant, wonderment. |
Source: compiled by the editor (additional references); see credits. | ||
Scrabble® Enable2K-Verified Anagrams | |
Direct Anagrams: detrainment. | |
| Words within the letters "a-d-e-e-i-m-n-n-r-t-t" | |
-1 letter: detainment, terminated. | |
-2 letters: detriment, entertain, entrained, interment, intreated, netminder, terminate. | |
-3 letters: admitter, antimere, attender, dementia, detainer, diameter, indenter, intender, interned, intranet, iterated, mannered, martinet, mattered, meditate, nattered, nitrated, rattened, remained, remanent, remanned, reminted, remitted, renitent, retained, retinted, tenanted, trainmen. | |
-4 letters: adenine, amender, antired, arenite, ariette, attired, demerit, dentate, dentine, detrain, dimeter, dinette, emerita, eminent. | |
| Words containing the letters "a-d-e-e-i-m-n-n-r-t-t" | |
+1 letter: determinants, detrainments. | |
+2 letters: determinantal, determination, indeterminate. | |
+3 letters: determinations, disheartenment, intermediating, intermediation, recontaminated. | |
+4 letters: codetermination, determinateness, disheartenments, indeterminately, indetermination, intermediations, redetermination, undemonstrative, underestimating, underestimation. | |
+5 letters: codeterminations, indeterminations, intermediateness, predetermination, redeterminations, underestimations. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
