Determinant

In linear algebra, the determinant is a function which associates a number to every square matrix, or alternatively, to every sequence of n vectors from Rⁿ.

It is used to calculate volumes in vector calculus, to characterize invertible matrices, and to explicitly describe the solution to a system of linear equations with Cramer's rule. The determinant of the square matrix A is denoted by det(A) or |A|.

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.

Definition

The following general definition was given by Leibniz and is known as the Leibniz formula:

$\det(A)=\sum _{\sigma \in S_{n}}\operatorname {sgn} (\sigma )\prod _{i=1}^{n}A_{i,\sigma (i)}$

Here, A_i,j stands for the entry in A in row number i and column number j. The sum is computed over all permutations of the numbers {1,...,n} (bijective functions σ : {1,...,n} -> {1,...,n}). 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. It is nevertheless useful for theoretical purposes; for instance, it shows that det is a polynomial map from R^n×n to R with total degree n.

Computing determinants

If A is a 1-by-1 matrix, then det(A) = A_1,1. If A is a 2-by-2 matrix, then det(A) = A_1,1 · A_2,2 - A_2,1 · A_1,2. For a 3-by-3 matrix A, the formula is more complicated:

det(A) = A_1,1·A_2,2·A_3,3 + A_1,3·A_3,2·A_2,1 + A_1,2·A_2,3·A_3,1

- A_3,1·A_2,2·A_1,3 - A_1,1·A_2,3·A_3,2 - A_1,2·A_2,1·A_3,3

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

• If A is a triangular matrix, i.e. A_i,j = 0 whenver i > j, then det(A) = A_1,1·A_2,2·...·A_n,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

$\det(A)=\sum _{j=1}^{n}\ (-1)^{i+j}\ A_{i,j}\ \det(A(i|j))$

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.

Interpretation of the determinant for real vectors

If viewed as a map that takes n vectors from Rⁿ and produces a real result, the determinant is multilinear, i.e. linear in each of its n entries. Furthermore, it is anti-symmetric, i.e. exchanging two arguments multiplies the determinant by -1.

The sign of the determinant of real vectors has a special significance because it serves to define the notion of orientation of coordinate systems. If three vectors in R³ are given, then they may be oriented similarly to the three vectors (1,0,0), (0,1,0), (0,0,1), i.e. similarly to the first three fingers of the right hand, in which case their determinant will be positive, or they may be oriented similarly to (1,0,0), (0,1,0), (0,0,-1), i.e. similarly to the first three fingers of the left hand, in which case their determinant will be negative. A similar statement holds true for higher dimensions.

The absolute value of the determinant of real vectors is important in volume computations because it is equal to the volume of the parallelepiped spanned by those vectors. This also means that the determinant of n vectors is zero if and only if the n vectors are linearly dependent. 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 -> R^m 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(A^tr * A)) × volume(S), where A^tr denotes the transpose of A.

One may also interpret the determinant of the square real n-by-n matrix A in terms of its eigenvalues. If λ₁,...,λ_n are the eigenvalues of A (some of which may be complex, and some may be repeated according to their multiplicity), then

det(A) = λ₁·λ₂·...·λ_n.

Compare this to the trace of A, tr(A), which equals the sum of the eigenvalues.

Compatibility with matrix multiplication and other properties

The determinant function is compatible with matrix multiplication in the following sense: if A and B are square matrices of the same size, then

det(AB) = det(A) · det(B).

Furthermore, A is invertible if and only if det(A) ≠ 0; if this is the case, then det(A^-1) = det(A)^-1.

As a consequence, if A and B are similar (in the sense that there exists an invertible matrix X such that A = X^-1BX), then det(A) = det(B). The converse is not true: there exist matrices with the same determinant which are not similar.

A matrix and its transpose have the same determinant:

det(A) = det(A^T).

Derivative

As we have seen, the determinant is a polynomial function from R^n×n to R, and as such is everwhere 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.

If f : V -> V is a linear transformation (also called "endomorphism") of a finite dimensional vector space over some field, we may define its determinant det(f) by first picking a basis of V, then representing f as a matrix with respect to that basis, and then computing the determinant of that matrix. This determinant will only depend on f and not on the basis chosen, because matrices for different bases are similar.

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 every entry.
• det is anti-symmetric, meaning that exchanging two arguments yields the negative determinant.
• det(e₁,..,e_n) = 1, where e_i is that element of M which has a 1 in the i-th coordinate and zeros elsewhere.