e-book The theory of determinants in the historical order of development: Volume 1

Such expressions are deducible from combinatorial arguments, Newton's identities , or the Faddeev—LeVerrier algorithm. In the general case, this may also be obtained from . The product and trace of such matrices are defined in a natural way as. An important arbitrary dimension n 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,. For a positive definite matrix A , the trace operator gives the following tight lower and upper bounds on the log determinant.

This relationship can be derived via the formula for the KL-divergence between two multivariate normal distributions. These inequalities can be proved by bringing the matrix A to the diagonal form. 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. This follows immediately by column expansion of the determinant, i. The rule is also implied by the identity.

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It has recently been shown that Cramer's rule can be implemented in O n 3 time,  which is comparable to more common methods of solving systems of linear equations, such as LU , QR , or singular value decomposition. This can be seen from the Leibniz formula , or from a decomposition like for the former case. When A is invertible , one has. When the blocks are square matrices of the same order further formulas hold. For example, if C and D commute i. If a block matrix is square, its characteristic polynomial can be factored with.

It can be seen, e. Its derivative can be expressed using Jacobi's formula : . In particular, if A is invertible, we have. This identity is used in describing the tangent space of certain matrix Lie groups. Indeed, repeatedly applying the above identities yields. The determinant is therefore also called a similarity invariant.

DETERMINANTS-1 Basic definition of & order of Determinant

The determinant of a linear transformation. By the similarity invariance, this determinant is independent of the choice of the basis for V and therefore only depends on the endomorphism T. A induces a linear map. This scalar coincides with the determinant of A , that is to say. This definition agrees with the more concrete coordinate-dependent definition. This follows from the characterization of the determinant given above.

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The vector space W of all alternating multilinear n -forms on an n -dimensional vector space V has dimension one. We call this scalar the determinant of T. This definition can also be extended where K is a commutative ring R , in which case a matrix is invertible if and only if its determinant is an invertible element in R. Such a matrix is called unimodular. Since it respects the multiplication in both groups, this map is a group homomorphism. The determinant respects these maps, i.

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. 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 the fact 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 bilinear form [ clarify ] with a regular element of R as value on some pair of arguments implies that R is commutative. 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.

Special Matrices

Manin matrices form the class closest to matrices with commutative elements. Determinants of matrices in superrings that is, Z 2 - graded rings are known as Berezinians or superdeterminants. The immanant generalizes both by introducing a character of the symmetric group S n in Leibniz's rule. Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra , where for applications like checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques.

Naive methods of implementing an algorithm to compute the determinant include using the Leibniz formula or Laplace's formula. Both these approaches are extremely inefficient for large matrices, though, since the number of required operations grows very quickly: it is of order n! For example, Leibniz's formula requires calculating n! Therefore, more involved techniques have been developed for calculating determinants.

Determinant - Wikipedia

Given a matrix A , some methods compute its determinant by writing A 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. These methods are of order O n 3 , which is a significant improvement over O n! The LU decomposition expresses A in terms of a lower triangular matrix L , an upper triangular matrix U and a permutation matrix P :.

The determinants of L and U can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of A is then. See determinant identities. Since the definition of the determinant does not need divisions, a question arises: do fast algorithms exist that do not need divisions? This is especially interesting for matrices over rings. Indeed, algorithms with run-time proportional to n 4 exist. An algorithm of Mahajan and Vinay, and Berkowitz  is based on closed ordered walks short clow.

It computes more products than the determinant definition requires, but some of these products cancel and the sum of these products can be computed more efficiently. The final algorithm looks very much like an iterated product of triangular matrices. Charles Dodgson i. Lewis Carroll of Alice's Adventures in Wonderland fame invented a method for computing determinants called Dodgson condensation. Unfortunately this interesting method does not always work in its original form.

Algorithms can also be assessed according to their bit complexity , i. For example, the Gaussian elimination or LU decomposition method is of order O n 3 , but the bit length of intermediate values can become exponentially long. Historically, determinants were used long 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 Europe, Cramer added to the theory, treating the subject in relation to sets of equations.

It was Vandermonde who first recognized determinants as independent functions. Immediately following, Lagrange 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 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 quantic. Gauss also arrived at the notion of reciprocal inverse determinants, and came very near the multiplication theorem.

On the same day November 30, 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. The next important figure was Jacobi  from He early used the functional determinant which Sylvester later called the Jacobian , and in his memoirs in Crelle's Journal for 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 and Cayley began their work. 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. As mentioned above, the determinant of a matrix with real or complex entries, say is zero if and only if the column vectors or the row vectors of the matrix are linearly dependent. Thus, determinants can be used to characterize linearly dependent vectors. 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. The determinant can be thought of as assigning a number to every sequence of n vectors in R n , by using the square matrix whose columns are the given vectors. For instance, an orthogonal matrix with entries in R n represents an orthonormal basis in Euclidean space. The determinant of such a matrix determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis.

As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. For a general differentiable function , much of the above carries over by considering the Jacobian matrix of f. The Jacobian also occurs in the inverse function theorem. In general, the n th-order Vandermonde determinant is . In general, the n th-order circulant determinant is . From Wikipedia, the free encyclopedia.

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