This article is about mathematics. For determinants in epidemiology, seeRisk factor. For determinants in immunology, seeEpitope.
Inmathematics, thedeterminant is ascalar-valuedfunction of the entries of asquare matrix. The determinant of a matrixA is commonly denoteddet(A),detA, or|A|. Its value characterizes some properties of the matrix and thelinear map represented, on a givenbasis, by the matrix. In particular, the determinant is nonzeroif and only if the matrix isinvertible and the corresponding linear map is anisomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse.
The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of atriangular matrix is the product of its diagonal entries.
The determinant of a2 × 2 matrix is
and the determinant of a3 × 3 matrix is
The determinant of ann ×n matrix can be defined in several equivalent ways, the most common beingLeibniz formula, which expresses the determinant as a sum of (thefactorial ofn) signed products of matrix entries. It can be computed by theLaplace expansion, which expresses the determinant as alinear combination of determinants of submatrices, or withGaussian elimination, which allows computing arow echelon form with the same determinant, equal to the product of the diagonal entries of the row echelon form.
Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on then ×n matrices that has the four following properties:
The exchange of two rows multiplies the determinant by−1.
Multiplying a row by a number multiplies the determinant by this number.
Adding a multiple of one row to another row does not change the determinant.
The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns.
The determinant is invariant undermatrix similarity. This implies that, given a linearendomorphism of afinite-dimensional vector space, the determinant of the matrix that represents it on abasis does not depend on the chosen basis. This allows defining thedeterminant of a linear endomorphism, which does not depend on the choice of acoordinate system.
The determinant has several key properties that can be proved by direct evaluation of the definition for-matrices, and that continue to hold for determinants of larger matrices. They are as follows:[1] first, the determinant of theidentity matrix is 1.Second, the determinant is zero if two rows are the same:
This holds similarly if the two columns are the same. Moreover,
Finally, if any column is multiplied by some number (i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:
The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.
If the matrix entries are real numbers, the matrixA represents thelinear map that maps thebasis vectors to the columns ofA. The images of the basis vectors form aparallelogram that represents the image of theunit square under the mapping. The parallelogram defined by the columns of the above matrix is the one with vertices at(0, 0),(a,c),(a +b,c +d), and(b,d), as shown in the accompanying diagram.
The absolute value ofad −bc is the area of the parallelogram, and thus represents the scale factor by which areas are transformed byA.
The absolute value of the determinant together with the sign becomes thesigned area of the parallelogram. The signed area is the same as the usualarea, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for theidentity matrix).
To show thatad −bc is the signed area, one may consider a matrix containing two vectorsu ≡ (a,c) andv ≡ (b,d) representing the parallelogram's sides. The signed area can be expressed as|u| |v| sinθ for the angleθ between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to thesine this already is the signed area, yet it may be expressed more conveniently using thecosine of the complementary angle to a perpendicular vector, e.g.u⊥ = (−c,a), so that|u⊥| |v| cosθ′ becomes the signed area in question, which can be determined by the pattern of thescalar product to be equal toad −bc according to the following equations:
Thus the determinant gives the area scale factor and the orientation induced by the mapping represented byA. When the determinant is equal to one, the linear mapping defined by the matrix preserves area and orientation.
The volume of thisparallelepiped is the absolute value of the determinant of the matrix formed by the columns constructed from the vectors r1, r2, and r3.
If ann ×nreal matrixA is written in terms of its column vectors, then
This means that maps the unitn-cube to then-dimensionalparallelotope defined by the vectors the region ( stands for "for all" as alogical symbol.)
The determinant gives thesignedn-dimensional volume of this parallelotope, and hence describes more generally then-dimensional volume scale factor of thelinear transformation produced byA.[2] (The sign shows whether the transformation preserves or reversesorientation.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fullyn-dimensional, which indicates that the dimension of the image ofA is less thann. Thismeans thatA produces a linear transformation which is neitheronto norone-to-one, and so is not invertible.
LetA be asquare matrix withn rows andn columns, so that it can be written as
The entries etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices whose entries are in acommutative ring.
The determinant ofA is denoted by det(A), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:
There are various equivalent ways to define the determinant of a square matrixA, i.e. one with the same number of rows and columns: the determinant can be defined via theLeibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question.
TheLeibniz formula for the determinant of a3 × 3 matrix is the following:
In this expression, each term has one factor from each row, all in different columns, arranged in increasing row order. For example,bdi hasb from the first row second column,d from the second row first column, andi from the third row third column. The signs are determined by how many transpositions of factors are necessary to arrange the factors in increasing order of their columns (given that the terms are arranged left-to-right in increasing row order): positive for an even number of transpositions and negative for an odd number. For the example ofbdi, the single transposition ofbd todb givesdbi, whose three factors are from the first, second and third columns respectively; this is an odd number of transpositions, so the term appears with negative sign.
Therule of Sarrus is a mnemonic for the expanded form of this determinant: the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the first two columns of the matrix are written beside it as in the illustration. This scheme for calculating the determinant of a3 × 3 matrix does not carry over into higher dimensions.
Generalizing the above to higher dimensions, the determinant of an matrix is an expression involvingpermutations and theirsignatures. A permutation of the set is abijective function from this set to itself, with values exhausting the entire set. The set of all such permutations, called thesymmetric group, is commonly denoted. The signature of a permutation is if the permutation can be obtained with an even number of transpositions (exchanges of two entries); otherwise, it is
Given a matrix
the Leibniz formula for its determinant is, usingsigma notation for the sum,
Usingpi notation for the product, this can be shortened into
.
TheLevi-Civita symbol is defined on then-tuples of integers in as0 if two of the integers are equal, and otherwise as the signature of the permutation defined by then-tuple of integers. With the Levi-Civita symbol, the Leibniz formula becomes
where the sum is taken over alln-tuples of integers in[3][4]
The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an matrixA as being composed of its columns, so denoted as
where thecolumn vector (for eachi) is composed of the entries of the matrix in thei-th column.
The determinant ismultilinear: if thejth column of a matrix is written as alinear combination of twocolumn vectorsv andw and a numberr, then the determinant ofA is expressible as a similar linear combination:
The determinant isalternating: whenever two columns of a matrix are identical, its determinant is 0:
If the determinant is defined using the Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any matrixA a number that satisfies these three properties.[5] This also shows that this more abstract approach to the determinant yields the same definition as the one using the Leibniz formula.
To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is astandard basis vector. These determinants are either 0 (by property 9) or else ±1 (by properties 1 and 12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear.[citation needed]
Interchanging any pair of columns of a matrix multiplies its determinant by −1. This follows from the determinant being multilinear and alternating (properties 2 and 3 above): This formula can be applied iteratively when several columns are swapped. For example Yet more generally, any permutation of the columns multiplies the determinant by thesign of the permutation.
If some column can be expressed as a linear combination of theother columns (i.e. the columns of the matrix form alinearly dependent set), the determinant is 0. As a special case, this includes: if some column is such that all its entries are zero, then the determinant of that matrix is 0.
Adding a scalar multiple of one column toanother column does not change the value of the determinant. This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0, since the determinant is alternating.
If is atriangular matrix, i.e., whenever or, alternatively, whenever, then its determinant equals the product of the diagonal entries: Indeed, such a matrix can be reduced, by appropriately adding multiples of the columns with fewer nonzero entries to those with more entries, to adiagonal matrix (without changing the determinant). For such a matrix, using the linearity in each column reduces to the identity matrix, in which case the stated formula holds by the very first characterizing property of determinants. Alternatively, this formula can also be deduced from the Leibniz formula, since the only permutation which gives a non-zero contribution is the identity permutation.
These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact,Gaussian elimination can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix using that method:
The determinant of thetranspose of equals the determinant ofA:
.
This can be proven by inspecting the Leibniz formula.[6] This implies that in all the properties mentioned above, the word "column" can be replaced by "row" throughout. For example, viewing ann ×n matrix as being composed ofn rows, the determinant is ann-linear function.
The determinant is amultiplicative map, i.e., for square matrices and of equal size, the determinant of amatrix product equals the product of their determinants:
This key fact can be proven by observing that, for a fixed matrix, both sides of the equation are alternating and multilinear as a function depending on the columns of. Moreover, they both take the value when is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.[7]
A matrix with entries in afield isinvertible precisely if its determinant is nonzero. This follows from the multiplicativity of the determinant and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by
.
In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size over a field) forms a group known as thegeneral linear group (respectively, asubgroup called thespecial linear group. More generally, the word "special" indicates the subgroup of anothermatrix group of matrices of determinant one. Examples include thespecial orthogonal group (which ifn is 2 or 3 consists of allrotation matrices), and thespecial unitary group.
Because the determinant respects multiplication and inverses, it is in fact agroup homomorphism from into the multiplicative group of nonzero elements of. This homomorphism is surjective and its kernel is (the matrices with determinant one). Hence, by thefirst isomorphism theorem, this shows that is anormal subgroup of, and that thequotient group is isomorphic to.
TheCauchy–Binet formula is a generalization of that product formula forrectangular matrices. This formula can also be recast as a multiplicative formula forcompound matrices whose entries are the determinants of all quadratic submatrices of a given matrix.[8][9]
Laplace expansion expresses the determinant of a matrixrecursively in terms of determinants of smaller matrices, known as itsminors. The minor is defined to be the determinant of the matrix that results from by removing the-th row and the-th column. The expression is known as acofactor. For every, one has the equality
which is called theLaplace expansion along theith row. For example, the Laplace expansion along the first row () gives the following formula:
Unwinding the determinants of these-matrices gives back the Leibniz formula mentioned above. Similarly, theLaplace expansion along the-th column is the equality
Laplace expansion can be used iteratively for computing determinants, but this approach is inefficient for large matrices. However, it is useful for computing the determinants of highly symmetric matrix such as theVandermonde matrixThen-term Laplace expansion along a row or column can begeneralized to write ann xn determinant as a sum ofterms, each the product of the determinant of ak xksubmatrix and the determinant of the complementary (n−k) x (n−k) submatrix.
The formula for the determinant of a matrix above continues to hold, under appropriate further assumptions, for ablock matrix, i.e., a matrix composed of four submatrices of dimension,, and, respectively. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving theSchur complement, is
If isinvertible, then it follows with results from the section on multiplicativity that
If the blocks are square matrices of thesame size further formulas hold. For example, if andcommute (i.e.,), then[11]
This formula has been generalized to matrices composed of more than blocks, again under appropriate commutativity conditions among the individual blocks.[12]
For and, the following formula holds (even if and do not commute).
Sylvester's determinant theorem states that forA, anm ×n matrix, andB, ann ×m matrix (so thatA andB have dimensions allowing them to be multiplied in either order forming a square matrix):
whereIm andIn are them ×m andn ×n identity matrices, respectively.
From this general result several consequences follow.
For the case of column vectorc and row vectorr, each withm components, the formula allows quick calculation of the determinant of a matrix that differs from the identity matrix by a matrix of rank 1:
More generally,[13] for any invertiblem ×m matrixX,
For a column and row vector as above:
For square matrices and of the same size, the matrices and have the same characteristic polynomials (hence the same eigenvalues).
A generalization is(seeMatrix determinant lemma), whereZ is anm ×m invertible matrix andW is ann ×n invertible matrix.
For the special case of matrices with complex entries, the determinant of the sum can be written in terms of determinants and traces in the following identity:
Properties of the determinant in relation to other notions
The determinant is closely related to two other central concepts in linear algebra, theeigenvalues and thecharacteristic polynomial of a matrix. Let be an matrix withcomplex entries. Then, by the Fundamental Theorem of Algebra, must have exactlyneigenvalues. (Here it is understood that an eigenvalue withalgebraic multiplicityμ occursμ times in this list.) Then, it turns out the determinant ofA is equal to theproduct of these eigenvalues,
The product of all non-zero eigenvalues is referred to aspseudo-determinant.
From this, one immediately sees that the determinant of a matrix is zero if and only if is an eigenvalue of. In other words, is invertible if and only if is not an eigenvalue of.
Here, is theindeterminate of the polynomial and is the identity matrix of the same size as. By means of this polynomial, determinants can be used to find theeigenvalues of the matrix: they are precisely theroots of this polynomial, i.e., those complex numbers such that
Thetrace tr(A) is by definition the sum of the diagonal entries ofA and also equals the sum of the eigenvalues. Thus, for complex matricesA,
or, for real matricesA,
Here exp(A) denotes thematrix exponential ofA, because every eigenvalueλ ofA corresponds to the eigenvalue exp(λ) of exp(A). In particular, given anylogarithm ofA, that is, any matrixL satisfying
the determinant ofA is given by
For example, forn = 2,n = 3, andn = 4, respectively,
In the general case, this may also be obtained from[19]
where the sum is taken over the set of all integerskl ≥ 0 satisfying the equation
The formula can be expressed in terms of the complete exponentialBell polynomial ofn argumentssl = −(l – 1)! tr(Al) as
This formula can also be used to find the determinant of a matrixAIJ with multidimensional indicesI = (i1,i2, ...,ir) andJ = (j1,j2, ...,jr). The product and trace of such matrices are defined in a natural way as
An important arbitrary dimensionn identity can be obtained from theMercator series expansion of the logarithm when the expansion converges. If every eigenvalue ofA is less than 1 in absolute value,
whereI is the identity matrix. More generally, if
is expanded as a formalpower series ins then all coefficients ofsm form >n are zero and the remaining polynomial isdet(I +sA).
These inequalities can be proved by expressing the traces and the determinant in terms of the eigenvalues. As such, they represent the well-known fact that theharmonic mean is less than thegeometric mean, which is less than thearithmetic mean, which is, in turn, less than theroot mean square.
The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is apolynomial function from to. In particular, it is everywheredifferentiable. Its derivative can be expressed usingJacobi's formula:[20]
where denotes theadjugate of. In particular, if is invertible, we have
Expressed in terms of the entries of, these are
Yet another equivalent formulation is
,
usingbig O notation. The special case where, the identity matrix, yields
This identity is used in describingLie algebras associated to certain matrixLie groups. For example, the special linear group is defined by the equation. The above formula shows that its Lie algebra is thespecial linear Lie algebra consisting of those matrices having trace zero.
Writing a matrix as where are column vectors of length 3, then the gradient over one of the three vectors may be written as thecross product of the other two:
Historically, determinants were used long before matrices: A determinant was originally defined as a property of asystem 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, determinants were first used in the Chinese mathematics textbookThe Nine Chapters on the Mathematical Art (九章算術, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed byCardano in 1545 by a determinant-like entity.[21]
Vandermonde (1771) first recognized determinants as independent functions.[23]Laplace (1772) gave the general method of expanding a determinant in terms of its complementaryminors: Vandermonde had already given a special case.[28] Immediately following,Lagrange (1773) treated determinants of the second and third order and applied it to questions ofelimination theory; he proved many special cases of general identities.
Gauss (1801) made the next advance. Like Lagrange, he made much use of determinants in thetheory of numbers. He introduced the word "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to thediscriminant of aquadratic form.[29] Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.[clarification needed]
The next contributor of importance isBinet (1811, 1812), who formally stated the theorem relating to the product of two matrices ofm columns andn rows, which for the special case ofm =n reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy,Cauchy also presented one on the subject. (SeeCauchy–Binet formula.) In this he used the word "determinant" in its present sense,[30][31] summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's.[23][32] With him begins the theory in its generality.
Jacobi (1841) used the functional determinant which Sylvester later called theJacobian.[33] In his memoirs inCrelle's Journal for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has calledalternants. About the time of Jacobi's last memoirs,Sylvester (1839) andCayley began their work.Cayley 1841 introduced the modern notation for the determinant using vertical bars.[34][35]
The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied byLebesgue,Hesse, and Sylvester;persymmetric determinants by Sylvester andHankel;circulants byCatalan,Spottiswoode,Glaisher, and Scott; skew determinants andPfaffians, in connection with the theory oforthogonal transformation, by Cayley; continuants by Sylvester;Wronskians (so called byMuir) byChristoffel andFrobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians andHessians by Sylvester; and symmetric gauche determinants byTrudi. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.
Determinants can be used to describe the solutions of alinear system of equations, written in matrix form as. This equation has a unique solution if and only if is nonzero. In this case, the solution is given byCramer's rule:
where is the matrix formed by replacing the-th column of by the column vector. This follows immediately by column expansion of the determinant, i.e.
where the vectors are the columns ofA. The rule is also implied by the identity
Cramer's rule can be implemented in time, which is comparable to more common methods of solving systems of linear equations, such asLU,QR, orsingular value decomposition.[36]
Determinants can be used to characterizelinearly dependent vectors: is zero if and only if the column vectors of the matrix are linearly dependent.[37] For example, given two linearly independent vectors, a third vector lies in theplanespanned by the former two vectors exactly if the determinant of the matrix consisting of the three vectors is zero. The same idea is also used in the theory ofdifferential equations: given functions (supposed to be timesdifferentiable), theWronskian is defined to be
It is non-zero (for some) in a specified interval if and only if the given functions and all their derivatives up to order are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case ofanalytic functions, this implies the given functions are linearly dependent. Seethe Wronskian and linear independence. Another such use of the determinant is theresultant, which gives a criterion when twopolynomials have a commonroot.[38]
The determinant can be thought of as assigning a number to everysequence ofn vectors inRn, by using the square matrix whose columns are the given vectors. The determinant will be nonzero if and only if the sequence of vectors is abasis forRn. In that case, the sign of the determinant determines whether theorientation of the basis is consistent with or opposite to the orientation of thestandard basis. In the case of an orthogonal basis, the magnitude of the determinant is equal to theproduct of the lengths of the basis vectors. For instance, anorthogonal matrix with entries inRn represents anorthonormal basis inEuclidean space, and hence has determinant of ±1 (since all the vectors have length 1). The determinant is +1 if and only if the basis has the same orientation. It is −1 if and only if the basis has the opposite orientation.
More generally, if the determinant ofA is positive,A represents an orientation-preservinglinear transformation (ifA is an orthogonal2 × 2 or3 × 3 matrix, this is arotation), while if it is negative,A switches the orientation of the basis.
As pointed out above, theabsolute value of the determinant of real vectors is equal to the volume of theparallelepiped spanned by those vectors. As a consequence, if is the linear map given by multiplication with a matrix, and is anymeasurablesubset, then the volume of is given by times the volume of.[39] More generally, if the linear map is represented by the matrix, then the-dimensional volume of is given by:
By calculating the volume of thetetrahedron bounded by four points, they can be used to identifyskew lines. The volume of any tetrahedron, given itsvertices,, or any other combination of pairs of vertices that form aspanning tree over the vertices.
A nonlinear map sends a small square (left, in red) to a distorted parallelogram (right, in red). The Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point (right, in translucent white), and the Jacobian determinant gives the ratio of the area of the approximating parallelogram to that of the original square.
the Jacobian matrix is then ×n matrix whose entries are given by thepartial derivatives
Its determinant, theJacobian determinant, appears in the higher-dimensional version ofintegration by substitution: for suitable functionsf and anopen subsetU ofRn (the domain off), the integral overf(U) of some other functionφ :Rn →Rm is given by
The above identities concerning the determinant of products and inverses of matrices imply thatsimilar matrices have the same determinant: two matricesA andB are similar, if there exists an invertible matrixX such thatA =X−1BX. Indeed, repeatedly applying the above identities yields
for some finite-dimensionalvector spaceV is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice ofbasis inV. By the similarity invariance, this determinant is independent of the choice of the basis forV and therefore only depends on the endomorphismT.
The above definition of the determinant using the Leibniz rule holds works more generally when the entries of the matrix are elements of acommutative ring, such as the integers, as opposed to thefield of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies still holds, as do all the properties that result from that characterization.[41]
A matrix is invertible (in the sense that there is an inverse matrix whose entries are in) if and only if its determinant is aninvertible element in.[42] For, this means that the determinant is +1 or −1. Such a matrix is calledunimodular.
Given aring homomorphism, there is a map given by replacing all entries in by their images under. The determinant respects these maps, i.e., the identity
holds. In other words, the displayed commutative diagram commutes.
For example, the determinant of thecomplex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo (the latter determinant being computed usingmodular arithmetic). In the language ofcategory theory, the determinant is anatural transformation between the two functors and.[43] Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism ofalgebraic groups, from the general linear group to themultiplicative group,
The determinant of a linear transformation of an-dimensional vector space or, more generally afree module of (finite)rank over a commutative ring can be formulated in a coordinate-free manner by considering the-thexterior power of.[44] The map induces a linear map
As is one-dimensional, the map is given by multiplying with some scalar, i.e., an element in. Some authors such as (Bourbaki 1998) use this fact todefine the determinant to be the element in satisfying the following identity (for all):
This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on-tuples of vectors in.For this reason, the highest non-zero exterior power (as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of and similarly for more involved objects such asvector bundles orchain complexes of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms with.[45]
The conventional definition of the determinant, as a sum over permutations over a product of matrix elements, can be written using the somewhat surprising notation of theBerezin integral. In this notation, the determinant can be written as
This holds for any-dimensional matrix The symbols are two-dimensional vectors of anti-commutingGrassmann numbers (aka "supernumbers"), taken from theGrassmann algebra. The here is theexponential function. The integral sign is meant to be understood as the Berezin integral. Despite the use of the integral symbol, this expression is in fact an entirely finite sum.
This unusual-looking expression can be understood as a notational trick that rewrites the conventional expression for the determinant
by using some novel notation. The anti-commuting property of the Grassmann numbers captures the sign (signature) of the permutation, while the integral combined with the ensures that all permutations are explored. That is, theTaylor's series for terminates after exactly terms, because the square of a Grassmann number is zero, and there are exactly distinct Grassmann variables. Meanwhile, the integral is defined to vanish, if the corresponding Grassmann number doesnot appear in the integrand. Thus, the integral selects out only those terms in the series that have exactly distinct variables; all lower-order terms vanish. Thus, the somewhat magical combination of the integral sign, the use of anti-commuting variables, and the Taylor's series for just encodes a finite sum, identical to the conventional summation.
This form is popular in physics, where it is often used as a stand-in for the Jacobian determinant. The appeal is that, notationally, the integral takes the form of apath integral, such as in thepath integral formulation for quantizedHamiltonian mechanics. An example can be found in the theory ofFadeev–Popov ghosts; although this theory may seem rather abstruse, it's best to keep in mind that the use of the ghost fields is little more than a notational trick to express a Jacobian determinant.
ThePfaffian of askew-symmetric matrix is the square-root of the determinant: that is, The Berezin integral form for the Pfaffian is even more suggestive; it is
The integrand has exactly the same formal structure as a normalGaussian distribution, albeit with Grassman numbers, instead of real numbers. This formal resemblance accounts for the occasional appearance of supernumbers in the theory ofstochastic dynamics andstochastic differential equations.
Determinants as treated above admit several variants: thepermanent of a matrix is defined as the determinant, except that the factors occurring in Leibniz's rule are omitted. Theimmanant generalizes both by introducing acharacter of thesymmetric group in Leibniz's rule.
This definition proceeds by establishing the characteristic polynomial independently of the determinant, and defining the determinant as the lowest order term of this polynomial. This general definition recovers the determinant for thematrix algebra, but also includes several further cases including the determinant of aquaternion,
For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated.Functional analysis provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators.
For operators in a finitefactor, one may define a positive real-valued determinant called theFuglede−Kadison determinant using the canonical trace. In fact, corresponding to everytracial state on avon Neumann algebra there is a notion of Fuglede−Kadison determinant.
For matrices over non-commutative rings, multilinearity and alternating properties are incompatible forn ≥ 2,[47] so there is no good definition of the determinant in this setting.
For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzerobilinear form[clarify] with aregular element ofR as value on some pair of arguments implies thatR is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notablyquasideterminants and theDieudonné determinant. For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include theq-determinant on quantum groups, theCapelli determinant on Capelli matrices, and theBerezinian onsupermatrices (i.e., matrices whose entries are elements of-graded rings).[48]Manin matrices form the class closest to matrices with commutative elements.
Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly innumerical linear algebra, where for applications such as checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques.[49]Computational geometry, however, does frequently use calculations related to determinants.[50]
While the determinant can be computed directly using the Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating (factorial) products for an matrix. Thus, the number of required operations grows very quickly: it isof order. The Laplace expansion is similarly inefficient. Therefore, more involved techniques have been developed for calculating determinants.
Gaussian elimination consists of left multiplying a matrix byelementary matrices for getting a matrix in arow echelon form. One can restrict the computation to elementary matrices of determinant1. In this case, the determinant of the resulting row echelon form equals the determinant of the initial matrix. As a row echelon form is atriangular matrix, its determinant is the product of the entries of its diagonal.
So, the determinant can be computed for almost free from the result of a Gaussian elimination.
Some methods compute by writing the matrix as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include theLU decomposition, theQR decomposition or theCholesky decomposition (forpositive definite matrices). These methods are of order, which is a significant improvement over.[51]
For example, LU decomposition expresses as a product
of apermutation matrix (which has exactly a single in each column, and otherwise zeros), a lower triangular matrix and an upper triangular matrix.The determinants of the two triangular matrices and can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of is just the sign of the corresponding permutation (which is for an even number of permutations and is for an odd number of permutations). Once such a LU decomposition is known for, its determinant is readily computed as
The order reached by decomposition methods has been improved by different methods. If two matrices of order can be multiplied in time, where for some, then there is an algorithm computing the determinant in time.[52] This means, for example, that an algorithm for computing the determinant exists based on theCoppersmith–Winograd algorithm. This exponent has been further lowered, as of 2016, to 2.373.[53]
In addition to the complexity of the algorithm, further criteria can be used to compare algorithms.Especially for applications concerning matrices over rings, algorithms that compute the determinant without any divisions exist. (By contrast, Gauss elimination requires divisions.) One such algorithm, having complexity is based on the following idea: one replaces permutations (as in the Leibniz rule) by so-calledclosed ordered walks, in which several items can be repeated. The resulting sum has more terms than in the Leibniz rule, but in the process several of these products can be reused, making it more efficient than naively computing with the Leibniz rule.[54] Algorithms can also be assessed according to theirbit complexity, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, theGaussian elimination (or LU decomposition) method is of order, but the bit length of intermediate values can become exponentially long.[55] By comparison, theBareiss Algorithm, is an exact-division method (so it does use division, but only in cases where these divisions can be performed without remainder) is of the same order, but the bit complexity is roughly the bit size of the original entries in the matrix times.[56]
If the determinant ofA and the inverse ofA have already been computed, thematrix determinant lemma allows rapid calculation of the determinant ofA +uvT, whereu andv are column vectors.
^A proof can be found in the Appendix B ofKondratyuk, L. A.; Krivoruchenko, M. I. (1992). "Superconducting quark matter in SU(2) color group".Zeitschrift für Physik A.344 (1):99–115.Bibcode:1992ZPhyA.344...99K.doi:10.1007/BF01291027.S2CID120467300.
^The first use of the word "determinant" in the modern sense appeared in: Cauchy, Augustin-Louis "Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et des signes contraires par suite des transpositions operées entre les variables qu'elles renferment," which was first read at the Institute de France in Paris on November 30, 1812, and which was subsequently published in theJournal de l'Ecole Polytechnique, Cahier 17, Tome 10, pages 29–112 (1815).
^In a non-commutative setting left-linearity (compatibility with left-multiplication by scalars) should be distinguished from right-linearity. Assuming linearity in the columns is taken to be left-linearity, one would have, for non-commuting scalarsa,b:a contradiction. There is no useful notion of multi-linear functions over a non-commutative ring.
^"... we mention that the determinant, though a convenient notion theoretically, rarely finds a useful role in numerical algorithms.", seeTrefethen & Bau III 1997, Lecture 1.
^Camarero, Cristóbal (2018-12-05). "Simple, Fast and Practicable Algorithms for Cholesky, LU and QR Decomposition Using Fast Rectangular Matrix Multiplication".arXiv:1812.02056 [cs.NA].
Cajori, Florian (1993),A history of mathematical notations: Including Vol. I. Notations in elementary mathematics; Vol. II. Notations mainly in higher mathematics, Reprint of the 1928 and 1929 originals, Dover,ISBN0-486-67766-4,MR3363427
