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Determinant

In linear algebra, the determinant is a function which associates a number to every square matrix. For instance, the 2-by-2 matrix

<math>A=\begin{bmatrix}a&b\\
c&d\end{bmatrix}</math> has determinant
<math>\det(A)=ad-bc</math>.
The formula for bigger matrices will be given below.

The determinant of A is also sometimes denoted by |A|, but this notation should be avoided as it is also used to denote other matrix functions, such as the square root of AA*.

Table of contents

History and applications

Historically, determinants were considered before matrices. Originally, a determinant was defined as a property of a system of linear equations. The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero). In this sense, two-by-two determinants were considered by Cardano at the end of the 16th century and larger ones by Leibniz about 100 years later.

Determinants are used to characterize invertible matrices, and to explicitly describe the solution to a system of linear equations with Cramer's rule. It can be used to find the eigenvalues of the matrix A through the characteristic polynomial p(x) = det(A-xIn).

One often thinks of the determinant as assigning a number to every sequence of n vectors in Rn, by using the square matrix whose columns are the given vectors. With this understanding, the sign of the determinant of a basis can be used to define the notion of orientation[?] in Euclidean spaces.

Determinants are used to calculate volumes in vector calculus: the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if the linear map f : Rn -> Rn is represented by the matrix A, and S is any measurable subset of Rn, then the volume of f(S) is given by |det(A)| × volume(S). More generally, if the linear map f : Rn -> Rm is represented by the m-by-n matrix A, and S is any measurable subset of Rn, then the n-dimensional volume of f(S) is given by √(det(ATA)) × volume(S).

Definition and Computation

Suppose A = (Ai,j) is a square matrix.

If A is a 1-by-1 matrix, then det(A) = A1,1. If A is a 2-by-2 matrix, then det(A) = A1,1 · A2,2 - A2,1 · A1,2. For a 3-by-3 matrix A, the formula is more complicated:

det(A) = A1,1·A2,2·A3,3 + A1,3·A3,2·A2,1 + A1,2·A2,3·A3,1
- A3,1·A2,2·A1,3 - A1,1·A2,3·A3,2 - A1,2·A2,1·A3,3

For a general n-by-n matrix, the determinant was defined by Gottfried Leibniz with what is now known as the Leibniz formula:

<math>\det(A) = \sum_{\sigma \in S_n}
\operatorname{sgn}(\sigma) \prod_{i=1}^n A_{i, \sigma(i)}</math> The sum is computed over all permutations σ of the numbers {1,...,n} and sgn(σ) denotes the signature of the permutation σ: +1 if σ is an even permutation and -1 if it is odd. See symmetric group for an explanation of even/odd permutations.

This formula contains n! summands and is therefore impractical to use if n is bigger than 3.

In general, determinants can be computed with the Gauss algorithm using the following rules:

  • If A is a triangular matrix, i.e. Ai,j = 0 whenver i > j, then det(A) = A1,1·A2,2·...·An,n
  • If B results from A by exchanging two rows or columns, then det(B) = - det(A)
  • If B results from A by multiplying one row or column with the number c, then det(B) = c · det(A)
  • If B results from A by adding a multiple of one row or column to another row or column, then det(B) = det(A).

Explicitly, starting out with some matrix, use the last three rules to convert it into a triangular matrix, then use the first rule to compute its determinant.

It is also possible to expand a determinant along a row or column using Laplace's formula. To do this along row i, say, we write

<math>\det(A) = \sum_{j=1}^n\ (-1)^{i+j}\ A_{i,j}\ \det(A(i|j))</math>

where A(i|j) denotes the n-1 by n-1 matrix resulting from A by removing the i-th row and j-th column.

Properties

The determinant is a multiplicative map in the sense that

det(AB) = det(A)det(B)     for all n-by-n matrices A and B

It is easy to see that det(rIn)=rn and thus

det(rA) = rn det(A)        for all n-by-n matrices A and all scalars r.

If A is invertible, then

det(A-1)=det(A)-1.

A matrix and its transpose have the same determinant:

det(A) = det(AT).

If A and B are similar, i.e. if there exists an invertible matrix X such that A = X-1BX, then by the multiplicative property,

det(A) = det(B).
Because of this, one may define the determinant of a linear map f : V -> V (where V is a finite-dimensional vector space) by choosing a basis for V, describing f as a matrix relative to this basis, and taking the determinant of this square matrix. The result will not depend on the basis chosen.

There exist matrices which have the same determinant but are not similar.

If A is a square n-by-n matrix with real or complex entries and if λ1,...,λn are the (complex) eigenvalues of A listed according to their algebraic multiplicities, then

det(A) = λ1·λ2·...·λn.
This follows from the fact that A is always similar to its Jordan form[?], an upper triangular matrix with the eigenvalues on the main diagonal.

From the connection between the determinant and the eigenvalues, one can derive a connection between the trace function, the exponential function, and the determinant:

det(exp(A)) = exp(tr(A)).

Derivative

The determinant of real square matrices is a polynomial function from Rn×n to R, and as such is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:

d det(A) = tr(adj(A) dA)
where adj(A) denotes the adjugate of A. In particular, if A is invertible, we have
d det(A) = det(A) tr(A-1 dA)
or, more colloquially,
det(A + X) - det(A) ≈ det(A) tr(A-1 X)
if the entries in the matrix X are sufficiently small. The special case where A is equal to the identity matrix I yields
det(I + X) ≈ 1 + tr(X).

Generalizations

It makes sense to define the determinant for matrices whose entries come from any commutative ring. The computation rules, the Leibniz formula and the compatibility with matrix multiplication remain valid, except that now a matrix A is invertible if and only if det(A) is an invertible element of the ground ring.

Abstractly, one may define the determinant as a certain anti-symmetric multilinear map as follows: if R is a commutative ring and M = Rn denotes the free R-module[?] with n generators, then

det : Mn -> R
is the unique map with the following properties:
  • det is R-linear in each of the n arguments.
  • det is anti-symmetric, meaning that if two of the n arguments are equal, then the determinant is zero.
  • det(e1,..,en) = 1, where ei is that element of M which has a 1 in the i-th coordinate and zeros elsewhere.

Linear Algebraists prefer to use the multilinear map approach to define determinant, whereas algebraists prefer to use the Leibniz formula.



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