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Wikipedia / Determinant

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix that has many properties which make it fundamental for the study of square matrices and linear transformations represented by them.

The determinant of a matrix is commonly denoted , , or . For by matrices with

There are several equivalent ways to define the determinant of an matrix. A simple and conceptual one is that it is the unique function that maps a product of matrices to a product of scalars and whose value on a triangular matrix is the product of the diagonal entries.

So, the determinant of the product of two square matrices is the product of their determinants. A square matrix is invertible if and only if its determinant has a multiplicative inverse. In the common case where the entries belong to a field, a matrix is invertible if and only if the determinant is nonzero. The adjugate matrix provides an explicit expression of the inverse matrix.

A consequence of the multiplicative property is that two similar matrices have the same determinant, and therefore the determinant of a linear transformation is well defined. For real linear transformations, the value of the determinant is the scale factor by which the transformation alters every volume in the space (positive if the orientation is preserved, negative if the orientation is reversed, zero if the transformation is not bijective).

The determinant is linear with respect to each row and each column of the matrix and is therefore a multilinear function. This means that it is a linear combination of the entries of each row or column of the matrix. Laplace expansion expresses the coefficients of this linear combination as determinants (of one dimension less), called cofactorss. Laplace expansion is useful for computing determinants when the rows or columns are very sparse, typically when there are only one or two nonzero entries in a row or column.

Determinants can be efficiently computed with Gaussian elimination and singular value decomposition. Determinants can also be theoretically computed by means of the Leibniz formula or Laplace expansion, but this is inefficient except in special cases.

Determinants occur throughout mathematics. Cramer's rule expresses the solution of a system of linear equations in terms of determinants. The characteristic polynomial of a square matrix, whose roots are the eigenvalues, is defined as a determinant with polynomial entries. In geometry, determinants are used to define or compute areass and volumess, as well as space orientation and colinearity. In analysis, the Jacobian determinant is used for change of variables in multiple integrals and its non-nullity is one of the hypotheses of the implicit function theorem in several variables. In algebraic geometry, the Hessian determinant allows finding inflexion points of an algebraic plane curve.

Two by two matrices

The determinant of a is

This simple form allows verifying by direct computation the properties listed in .

Parallelogram area

One of the main uses of determinants is the computation of the area of the parallelogram defined by two vectors in a Euclidean plane.

In fact, the two column vectors define a parallelogram with vertices , , , and . The area of this parallogram is the absolute value of the determinant of the matrix. The sign of the determinant determines the relative orientation of the vectors: the determinant is positive if one passes by a counterclockwise rotation from the direction of the vector to that of, and negative if a clockwise rotation is needed (if the vectors are aligned, the determinant is zero). So, the determinant is the signed area of the parallelogram.

The proof that the determinant is the signed area of the parallelogram can be done either with trigonometry or by dissection and rearrangement, as shown in the figure.

Definitions

The determinant of a square matrix is a scalar uniquely associated to it. Commonly, matrix entries and determinants are real or complex numbers, but, more generally, matrix entries may belong to any commutative ring is a square matrix with rows and columns, its determinant is denoted ,

There are various equivalent ways to define the determinant of a square matrix , and it depends on authors which definition is chosen as the primary one. In this article, the definition by the multiplicative property has been chosen as the primary one, because it is simple and does not require specific background.

By multiplicative property

The determinant is the unique function from the square matrices to the scalars that maps a product of matrices of the same size to the product of their determinants, and maps every triangular matrix to the product of their diagonal entries

Such a definition requires the proof that such a function exists and is unique. This will be a byproduct of the proof of the equivalence with other definitions.

By elementary transformations

An elementary row transformation is a transformation of a matrix of one of the following types, where denotes the th row of the matrix:

  1. Multiplying one row by a scalar:

  2. Adding to a row a scalar multiple of another row:

  3. Exchanging two rows:

Elementary column transformations are defiend similarly by replacing "row" with "column". Transformations of type 3 may be obtained as a succession of thansformations of types 1 and 2: the exchange of rows and may be done by the following succession of transformations, where denotes the current

This leads to another definition of determinants:

  1. The determinant is the unique function of the entries of a square matrix such that the determinant of the identity matrix is 1, the transformations of type 1 multiply the determinant by the involved scalar, and the transformations of type 2 do not change thedeterminant; it follows that the transformations of type 3 multiply the determinant by –1.

For proving that this definition is equivalent to the preceding one, one has to interpret elementary transformations in terms of elementary matrices. In fact, an elementaty transformation amounts to left-multiply the matrix by the diagonal matrix with all diagonal elements equal to except the th that equals . An elementary transformation amounts ro left-multiply the matrix by a triangular matrix in which all diagonal entries are equal to , the entry in the th row and the th column is equal to , and all other entries are zero.

A function determinant that satisfies the definition by multiplicative properties satisfy the definition by elementary transformations since the identity matix is a triangular matrix and the other conditions of the latter definition are the restriction of the multiplicative property to the case where the left factor is an elementary matrix.

Conversely, a determinant function that satisfies the definition by elementary transformations satisfies also the other definition. Indeed, Gaussian elimination allows proving that every matrix can be expressed as a product of elementary matrices. So, for proving , it duffices to express as a product of elementary matrices and apply associative property of matrix product. Uniqueness of the determinant results also from Gaussian elimination. Indeed it consists to transform, by means of elementary transformations, any matrix into a row echelon form, which is a specific triangular matrix. This allows computing the determinant and thus proving it uniqueness. However, this does not prove the existence of the determinant, since this would require a proof that different choices for the row reduction give the same result.

By Leibniz formula

Leibniz formula is one of the most common definitions of determinants. It is an explicit expression of a determinant as a multivariate polynomial function of the matrix entries. So, the proof that is satisfies the preceding definitions implies the existence proof that they require. Also, per uniqueness, this proves also the equivalence with the other definitions.

To write Leibniz formula, one needs permutations and their signatures where runs over all permutations of .

When is the empty matrix, the usual conventions for the empty set and the empty product give . If ,one has

As an alternating multilinear form

matrix A columns, so denoted as

where the column vector (for each i) is composed of the entries of the matrix in the i-th column.

  1. is an identity matrix.

  2. The determinant is multilinear: if the j is written as a linear combination of two column vectors v and w and a number r, then the determinant of A is expressible as a similar linear combination:

  3. The determinant is alternating: whenever two columns of a matrix are identical, its determinant is 0:

matrix A a number that satisfies these three properties. 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 a standard basis vector. These determinants are either 0 (if the columns are linearly dependent, by property 3) or else ±1 (by property 1 and 3 - the minus sign appears when the columns are permuted according to an odd permutation), 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.

Exterior algebra and the determinant of n vectors

.

matrix whose columns are

such that

.

Expanding by multilinearity gives

By multilinearity,

.

Leibniz formula

, and

Consequently,

which is the usual Leibniz formula for the determinant.

Row formulation

are regarded as covectors

, then

Therefore, the determinant may be defined using either the wedge product of the columns or the wedge product of the rows, although the rows naturally belong to the dual space.

Examples

3 × 3 matrices

The Leibniz formula for the determinant of a 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 has b from the first row second column, d from the second row first column, and i 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 of bdi, the single transposition of bd to db gives dbi, 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. The rule 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 a matrix does not carry over into higher dimensions.

n × n matrices

matrix is an expression involving permutations and their signatures is a bijective function exhausting the entire set. The set of all such permutations, called the symmetric group

Given a matrix

the Leibniz formula for its determinant is, using sigma notation for the sum,

Using pi notation for the product, this can be shortened into

  1. .

The Levi-Civita symbol is defined on the -tuples as if two of the integers are equal, and otherwise as the signature of the permutation defined by the n-tuple of integers. With the Levi-Civita symbol, the Leibniz formula becomes

where the sum is taken over all

Properties

Immediate consequences

These rules have several further consequences:

  • The determinant is a homogeneous function).

  • Interchanging any pair of columns of a matrix multiplies its determinant by (Given property 2 above, this is equivalent to property 3, which is why property 3 uses the term alternating In other words, any permutation of the columns multiplies the determinant by the sign of the permutation.

  • If any column can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a linearly dependent set), the determinant is 0. As a special case, if any column has all zero entries, then the determinant is 0.

  • Adding a scalar multiple of one column to another column does not change the value of the determinant. This is a consequence of multilinearity and being alternating: 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.

  • is a triangular matrix Indeed, such a matrix can be reduced, by appropriately adding multiples of the columns with fewer nonzero entries to those with more entries, to a diagonal matrix which gives a non-zero contribution is the identity permutation.

Example

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 using that method:

Transpose

The determinant of the transpose equals the determinant of A:

  1. .

This can be proven by inspecting the Leibniz formula. This implies that in all the properties mentioned above, the word "column" can be replaced by "row" throughout. For example, viewing an matrix as being composed of n rows, the determinant is an n-linear function.

Multiplicativity and matrix groups

The determinant is a multiplicative map of equal size, the determinant of a matrix product equals the product of their determinants:

is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.

with entries in a field is invertible 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

  1. .

) forms a group known as the general linear group (respectively, a subgroup called the special linear group. More generally, the word "special" indicates the subgroup of another matrix group of matrices of determinant one. Examples include the special orthogonal group (which if n is 2 or 3 consists of all rotation matrices), and the special unitary group.

Because the determinant respects multiplication and inverses, it is in fact a group homomorphism (the matrices with determinant one). Hence, by the first isomorphism theorem is a normal subgroup, and that the quotient group.

The Cauchy–Binet formula is a generalization of that product formula for rectangular matrices. This formula can also be recast as a multiplicative formula for compound matrices whose entries are the determinants of all quadratic submatrices of a given matrix.

Laplace expansion

Laplace expansion recursively in terms of determinants of smaller matrices, known as its minors is known as a cofactor, one has the equality

which is called the Laplace expansion along the th row) gives the following formula:

-matrices gives back the Leibniz formula mentioned above. Similarly, 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 the Vandermonde matrixThe n-term Laplace expansion along a row or column can be generalized to write an n x n terms, each the product of the determinant of a k x k submatrix and the determinant of the complementary (n−k) x (n−k) submatrix.

Adjugate matrix

The adjugate matrix is the transpose of the matrix of the cofactors, that is,

For every matrix, one has

Thus the adjugate matrix can be used for expressing the inverse of a nonsingular matrix:

Block matrices

matrix above continues to hold, under appropriate further assumptions, for a block matrix, respectively. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the Schur complement, is

is invertible, then it follows with results from the section on multiplicativity that

matrix.

is invertible, namely

Both results can be combined to derive Sylvester's determinant theorem, which is also stated below.

If the blocks are square matrices of the same commute), then

blocks, again under appropriate commutativity conditions among the individual blocks.

do not commute).

It is possible to compute the determinant by the block matrices in a fast way with the use of fast matrix multiplication algorithms decomposition.

Sylvester's determinant theorem

Sylvester's determinant theorem states that for A, an matrix, and B, an matrix (so that A and B have dimensions allowing them to be multiplied in either order forming a square matrix):

where Im and In are the and identity matrices, respectively.

From this general result several consequences follow.

(see Matrix determinant lemma), where Z is an invertible matrix and W is an invertible matrix.

Sum

of two square matrices of the same size is not in general expressible in terms of the determinants of A and of B.

However, for positive semidefinite matrices with the corollary

Brunn–Minkowski theorem implies that the th root of determinant is a concave function, when restricted to Hermitian matrices. Therefore, if and since the th root of the determinant is a homogeneous function.

Sum identity for 2×2 matrices

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

Eigenvalues and characteristic polynomial

The determinant is closely related to two other central concepts in linear algebra, the eigenvalues and the characteristic polynomial matrix with complex must have exactly n eigenvalues. (Here it is understood that an eigenvalue with algebraic multiplicity occurs times in this list.) Then, it turns out the determinant of is equal to the product of these eigenvalues,

The product of all non-zero eigenvalues is referred to as pseudo-determinant.

.

The characteristic polynomial is defined as

is the indeterminate. By means of this polynomial, determinants can be used to find the eigenvalues: they are precisely the roots such that

A Hermitian matrix is positive definite if all its eigenvalues are positive. Sylvester's criterion asserts that this is equivalent to the determinants of the submatrices

.

Trace

The trace tr(A) is by definition the sum of the diagonal entries of and also equals the sum of the eigenvalues. Thus, for complex matrices ,

or, for real matrices ,

Here exp() denotes the matrix exponential of , because every eigenvalue of corresponds to the eigenvalue exp() of exp(). In particular, given any logarithm of , that is, any matrix satisfying

the determinant of is given by

For example, for , , and , respectively,

cf. Cayley-Hamilton theorem. Such expressions are deducible from combinatorial arguments, Newton's identities, or the Faddeev–LeVerrier algorithm. That is, for generic , the signed constant term of the characteristic polynomial, determined recursively from

In the general case, this may also be obtained from

where the sum is taken over the set of all integers satisfying the equation

The formula can be expressed in terms of the complete exponential Bell polynomial of n arguments sl = −(l – 1)! tr(Al) as

This formula can also be used to find the determinant of a matrix with multidimensional indices and . The product and trace of such matrices are defined in a natural way as

An important arbitrary dimension identity can be obtained from the Mercator series expansion of the logarithm when the expansion converges. If every eigenvalue of A is less than 1 in absolute value,

where is the identity matrix. More generally, if

is expanded as a formal power series in then all coefficients of for are zero and the remaining polynomial is .

Upper and lower bounds

For a positive definite matrix , the trace operator gives the following tight lower and upper bounds on the log determinant

with equality if and only if . This relationship can be derived via the formula for the Kullback–Leibler divergence between two multivariate normal distributions.

Also,

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 the harmonic mean is less than the geometric mean, which is less than the arithmetic mean, which is, in turn, less than the root mean square.

Derivative

The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a polynomial. In particular, it is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:

denotes the adjugate is invertible, we have

, these are

Yet another equivalent formulation is

  1. ,

using big O notation, the identity matrix, yields

This identity is used in describing Lie algebras associated to certain matrix Lie groups. The above formula shows that its Lie algebra is the special linear Lie algebra consisting of those matrices having trace zero.

are column vectors of length 3, then the gradient over one of the three vectors may be written as the cross product of the other two:

History

Historically, determinants were used long before matrices: A determinant was originally 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 Europe, solutions of linear systems of two equations were expressed by Cardano in 1545 by a determinant-like entity.

Determinants proper originated separately from the work of Seki Takakazu in 1683 in Japan and parallelly of Leibniz in 1693. stated, without proof, Cramer's rule. Both Cramer and also were led to determinants by the question of plane curves passing through a given set of points.

Vandermonde (1771) first recognized determinants as independent functions. gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case. Immediately following, Lagrange (1773) treated determinants of the second and third order and applied it to questions of elimination theory; he proved many special cases of general identities.

Gauss (1801) made the next advance. Like Lagrange, he made much use of determinants in the theory of numbers. He introduced the word "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to the discriminant of a quadratic form. Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.

The next contributor of importance is Binet (1811, 1812), who formally stated the theorem relating to the product of two matrices of m columns and n rows, which for the special case of 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. (See Cauchy–Binet formula.) In this he used the word "determinant" in its present sense, 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. With him begins the theory in its generality.

used the functional determinant which Sylvester later called the Jacobian. In his memoirs in Crelle's Journal for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called alternants. About the time of Jacobi's last memoirs, Sylvester (1839) and Cayley began their work. introduced the modern notation for the determinant using vertical bars.

The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by Lebesgue, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hankel; circulants by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians (so called by Muir) by Christoffel and Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and Hessians by Sylvester; and symmetric gauche determinants by Trudi. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.

Recent history

A coordinate-free approach to the definition of determinants was the result of work by Grassmann, Bourbaki and Lang.

Applications

Cramer's rule

Determinants can be used to describe the solutions of a linear system of equations is nonzero. In this case, the solution is given by Cramer's rule:

. This follows immediately by column expansion of the determinant, i.e.

are the columns of A. The rule is also implied by the identity

time, which is comparable to more common methods of solving systems of linear equations, such as LU, QR, or singular value decomposition.

Linear independence

Determinants can be used to characterize linearly dependent are linearly dependent. lies in the plane spanned matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations times differentiable), the Wronskian is defined to be

are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence. Another such use of the determinant is the resultant, which gives a criterion when two polynomials have a common root.

Cross Product

:

From this definition, one can easily derive the anticommutative is equivalent to swapping two rows in the determinant, which entails changing the sign of the determinant, giving:

  1. .

Orientation of a basis

The determinant can be thought of as assigning a number to every sequence of n vectors in Rn, 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 a basis for Rn. In that case, the sign of the determinant determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis. In the case of an orthogonal basis, the magnitude of the determinant is equal to the product of the lengths of the basis vectors. For instance, an orthogonal matrix with entries in Rn represents an orthonormal basis in Euclidean 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 of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal or matrix, this is a rotation), while if it is negative, A switches the orientation of the basis.

Signed area of a parallelotope

If an real matrix A, then

maps the unit n-cube to the n-dimensional parallelotope stands for "for all" as a logical symbol.)

The determinant gives the signed n and hence describes more generally the n-dimensional volume scale factor of the linear transformation produced by A. (The sign shows whether the transformation preserves or reverses orientation.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully n-dimensional, which indicates that the dimension of the image of A is less than n. This means that A produces a linear transformation which is neither onto nor one-to-one, and so is not invertible.

Volume and Jacobian determinant

As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped is any measurable subset.-dimensional is given by:

this is zero.

By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. The volume of any tetrahedron, given its vertices, or any other combination of pairs of vertices that form a spanning tree over the vertices.

For a general differentiable function, much of the above carries over by considering the Jacobian matrix of f. For

the Jacobian matrix is the matrix whose entries are given by the partial derivatives

Its determinant, the Jacobian determinant, appears in the higher-dimensional version of integration by substitution: for suitable functions f and an open subset U of Rn (the domain of f), the integral over f(U) of some other function is given by

The Jacobian also occurs in the inverse function theorem.

When applied to the field of Cartography, the determinant can be used to measure the rate of expansion of a map near the poles.

Areas and Collinearity

is given by:

is chosen to yield a positive area. An immediate corollary of this result is that if the determinant is zero, there is no triangle, which implies the points are collinear. The determinant, therefore, provides a useful tool to test for collinearity.

as:

Abstract algebraic aspects Abstract formulation

Determinant of an endomorphism

The above identities concerning the determinant of products and inverses of matrices imply that similar matrices have the same determinant: two matrices A and B are similar, if there exists an invertible matrix X such that . Indeed, repeatedly applying the above identities yields

The determinant is therefore also called a similarity invariant. The determinant of a linear transformation

for some finite-dimensional vector space V is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of basis in V. By the similarity invariance, this determinant is independent of the choice of the basis for V and therefore only depends on the endomorphism T.

Square matrices over commutative rings

The above definition of the determinant using the Leibniz rule works more generally when the entries of the matrix are elements of a commutative ring, as opposed to the field still holds, as do all the properties that result from that characterization.

) if and only if its determinant is an invertible element., this means that the determinant is +1 or −1. Such a matrix is called unimodular.

The determinant being multiplicative, it defines a group homomorphism

between the general linear group) and the multiplicative group. Since it respects the multiplication in both groups, this map is a group homomorphism.

Given a ring homomorphism. The determinant respects these maps, i.e., the identity

holds. In other words, the displayed commutative diagram commutes.

For example, the determinant of the complex conjugate (the latter determinant being computed using modular arithmetic). In the language of category theory, the determinant is a natural transformation. Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of algebraic groups, from the general linear group to the multiplicative group,

Exterior algebra

or, more generally a free module of (finite) rank-th exterior power. induces a linear map

. Some authors such as use this fact to define):

and similarly for more involved objects such as vector bundles or chain complexes.

Berezin integral

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 the Berezin integral. In this notation, the determinant can be written as

-dimensional vectors of anti-commuting Grassmann numbers (aka "supernumbers"), taken from the Grassmann algebra here is the exponential 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

ensures that all permutations are explored. That is, the Taylor's series distinct Grassmann variables. Meanwhile, the integral is defined to vanish, if the corresponding Grassmann number does not 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 a path integral, such as in the path integral formulation for quantized Hamiltonian mechanics. An example can be found in the theory of Fadeev–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.

The Pfaffian of a skew-symmetric matrix The Berezin integral form for the Pfaffian is even more suggestive; it is

The integrand has exactly the same formal structure as a normal Gaussian distribution, albeit with Grassman numbers, instead of real numbers. This formal resemblance accounts for the occasional appearance of supernumbers in the theory of stochastic dynamics and stochastic differential equations.

Determinants as treated above admit several variants: the permanent occurring in Leibniz's rule are omitted. The immanant generalizes both by introducing a character of the symmetric group in Leibniz's rule.

Determinants for finite-dimensional algebras

For any associative algebra that is finite-dimensional, there is a determinant map

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 the matrix algebra, but also includes several further cases including the determinant of a quaternion,

  1. ,

the norm of a field extension, as well as the Pfaffian of a skew-symmetric matrix and the reduced norm of a central simple algebra.

Infinite matrices

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.

The Fredholm determinant defines the determinant for operators known as trace class operators by an appropriate generalization of the formula

Another infinite-dimensional notion of determinant is the functional determinant.

Operators in von Neumann algebras

For operators in a finite factor, one may define a positive real-valued determinant called the Fuglede−Kadison determinant using the canonical trace. In fact, corresponding to every tracial state on a von Neumann algebra there is a notion of Fuglede−Kadison determinant.

Related notions for non-commutative rings

For matrices over non-commutative rings, multilinearity and alternating properties are incompatible for , 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 nonzero with a regular element of R as value on some pair of arguments implies that R is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably quasideterminants and the Dieudonné 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 the q-determinant on quantum groups, the Capelli determinant on Capelli matrices, and the Berezinian on supermatrices-graded rings). Manin matrices form the class closest to matrices with commutative elements.

Calculation

Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for applications such as checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques. Computational geometry, however, does frequently use calculations related to determinants.

factorial matrix. Thus, the number of required operations grows very quickly: it is of order. The Laplace expansion is similarly inefficient. Therefore, more involved techniques have been developed for calculating determinants.

Gaussian elimination

Gaussian elimination consists of left multiplying a matrix by elementary matrices for getting a matrix in a row echelon form. One can restrict the computation to elementary matrices of determinant . In this case, the determinant of the resulting row echelon form equals the determinant of the initial matrix. As a row echelon form is a triangular 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.

Decomposition methods

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 the LU decomposition, the QR decomposition or the Cholesky decomposition (for positive definite matrices.

as a product

of a permutation matrix, its determinant is readily computed as

Further methods

can be multiplied. algorithm for computing the determinant exists based on the Coppersmith–Winograd algorithm. This exponent has been further lowered, as of 2016, to 2.373.

is based on the following idea: one replaces permutations (as in the Leibniz rule) by so-called closed 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. Algorithms can also be assessed according to their bit complexity, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, the Gaussian elimination, but the bit length of intermediate values can become exponentially long. By comparison, the Bareiss Algorithm.

If the determinant of A and the inverse of A have already been computed, the matrix determinant lemma allows rapid calculation of the determinant of , where u and v are column vectors.

Charles Dodgson (i.e. Lewis Carroll of Alice's Adventures in Wonderland fame) invented a method for computing determinants called Dodgson condensation. This method does not always work in its original form.

See also

Notes

Citations

References

Historical references