In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation. Determinants are important both in calculus, where they enter the substitution rule for several variables, and in multilinear algebra.
For a fixed nonnegative integer n, there is a unique determinant function for the n×n matrices over any commutative ring R. In particular, this function exists when R is the field of real or complex numbers.
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The determinant of a matrix A is also sometimes denoted by |A|. This notation can be ambiguous since it is also used for certain matrix norms and for the absolute value. However, often the matrix norm will be denoted with double vertical bars (e.g., ||A||) and may carry a subscript as well. Thus, the vertical bar notation for determinant is frequently used (e.g., Cramer's rule and minors).
For example, for matrix
the determinant might be indicated by
or more explicitly as
That is, the square braces around the matrices are replaced with elongated vertical bars.
The 2×2 matrix,
has determinant
A matrix with real number entries also produces a representation of the oriented area of the parallelogram with vertices at (0,0), (a,b), (a + c, b + d), and (c,d). The oriented area is the same as the usual area, except that it is negative when the vertices are listed in clockwise order.
The assumption here is that a linear transformation is applied to row vectors as the vector-matrix product , where
is a column vector. The parallelogram in the figure is obtained by multiplying matrix A (which stores the co-ordinates of our parallelogram) with each of the row vectors
and
in turn. These row vectors define the vertices of the unit square. With the more common matrix-vector product
the parallelogram has vertices at
and
(note that
).
Thus when the determinant is equal to one, then the matrix represents an equi-areal mapping.
The 3×3 matrix:
Using the cofactor expansion on the first row of the matrix we get:
which can be remembered, by applying the Rule of Sarrus, as 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 below:
Note that this mnemonic does not carry over into higher dimensions.
Determinants are used to characterize invertible matrices (i.e., a matrix is invertible if and only if it has a non-zero determinant), and to explicitly describe the solution to a system of linear equations with Cramer's rule. This rule is denoted
where the matrix is the matrix formed by the insertion of the solution column vector.
They can be used to find the eigenvalues of the matrix through the characteristic polynomial
where I is the identity matrix of the same dimension as A.
One often thinks of the determinant as assigning a number to every sequence of vectors in
, 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. The determinant of a set of vectors is positive if the vectors form a right-handed coordinate system, and negative if left-handed.
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 is represented by the matrix
, and
is any measurable subset of
, then the volume of
is given by
. More generally, if the linear map
is represented by the
-by-
matrix
, and
is any measurable subset of
, then the
-dimensional volume of
is given by
. 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 a, b, c, and d, is (1/6)·|det(a − b, b − c, c − d)|, or any other combination of pairs of vertices that form a simply connected graph.
The formal extension to arbitrary dimensions was made by Levi-Civita, using a pseudo-tensor symbol (Levi-Civita symbol). A definition of the determinant can be obtained by considering the following result.
Theorem. Let Mn(K) denote the set of all matrices over the field K. There exists exactly one function
with the two properties:
One can then define the determinant as the unique function with the above properties.
In proving the above theorem, one also obtains the Leibniz formula:
Here the sum is computed over all permutations of the numbers {1,2,...,n}.
denotes the set of all n! permutations of the set S = {1,2,...,n}.
denotes the signature of the permutation
: +1 if
is an even permutation and −1 if it is odd.
can also denote the signature of the number of inversions of the product of the permutation which is the approach used in some textbooks.
This formula contains (factorial) summands, and it is therefore impractical to use it to calculate determinants for large
.
For small matrices, one obtains the following formulas:
It is also possible to consider 0-by-0 matrices. There is only one 0-by-0 matrix and its determinant is one.[1]
In general, determinants can be computed using Gaussian elimination using the following rules:
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, which is efficient for relatively small matrices. To do this along row , say, we write
where the represent the matrix cofactors, i.e.
is
times the minor
, which is the determinant of the matrix that results from
by removing the
-th row and the
-th column, and
is the length of the matrix.
Suppose we want to compute the determinant of
We can go ahead and use the Leibniz formula directly:
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Alternatively, we can use Laplace's formula to expand the determinant along a row or column. It is best to choose a row or column with many zeros, so we will expand along the second column:
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A third way (and the method of choice for larger matrices) would involve the Gauss algorithm. When doing computations by hand, one can often shorten things dramatically by cleverly adding multiples of columns or rows to other columns or rows; this does not change the value of the determinant, but may create zero entries which simplifies the subsequent calculations. In this example, adding the second column to the first one is especially useful:
and this determinant can be quickly expanded along the first column:
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The determinant is a multiplicative map in the sense that
This is generalized by the Cauchy-Binet formula to products of non-square matrices.
It is easy to see that and thus
A matrix over a commutative ring R is invertible if and only if its determinant is a unit in R. In particular, if A is a matrix over a field such as the real or complex numbers, then A is invertible if and only if det(A) is not zero. In this case we have
Expressed differently: the vectors v1,...,vn in Rn form a basis if and only if det(v1,...,vn) is non-zero.
A matrix and its transpose have the same determinant:
The determinants of a complex matrix and of its conjugate transpose are conjugate:
(Note the conjugate transpose is identical to the transpose for a real matrix)
The determinant of a matrix exhibits the following properties under elementary matrix transformations of
:
This follows from the multiplicative property and the determinants of the elementary matrix transformation matrices.
If and
are similar, i.e., if there exists an invertible matrix
such that
=
, then by the multiplicative property,
This means that the determinant is a similarity invariant. Because of this, the determinant of some linear transformation T : V → V for some finite dimensional vector space V is independent of the basis for V. The relationship is one-way, however: there exist matrices which have the same determinant but are not similar.
If is a square
-by-
matrix with real or complex entries and if λ1,...,λn are the (complex) eigenvalues of
listed according to their algebraic multiplicities, then
This follows from the fact that is always similar to its Jordan normal form, an upper triangular matrix with the eigenvalues on the main diagonal.
Sylvester's determinant theorem states that for any m-by-n matrices A and B,
For the case of (column) vectors a and b, this equality becomes
With X a nonsingular m-by-m matrix, this last expression generalizes to
Suppose, are
matrices respectively. Then
This can be seen from the Leibniz formula or by induction on n. Employing the following identity
leads to
A similar identity with factored out can be derived analogously.[3]
If are diagonal matrices, then
From this connection between the determinant and the eigenvalues, one can derive a connection between the trace function, the exponential function, and the determinant:
Performing the substitution in the above equation yields
which is closely related to the Fredholm determinant. Similarly,
For n-by-n matrices there are the relationships:
which are closely related to Newton's identities.
The determinant of real square matrices is a polynomial function from to
, and as such is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:
where adj(A) denotes the adjugate of A. In particular, if A is invertible, we have
In component form, these are
When is a small number these are equivalent to
The special case where is equal to the identity matrix
yields
A useful property in the case of 3 x 3 matrices is the following:
A may be written as where
,
,
are vectors, then the gradient over one of the three vectors may be written as the cross product of the other two:
An n × n square matrix A may be thought of as the coordinate representation of a linear transformation of an n-dimensional vector space V. Given any linear transformation
we can define the determinant of A as the determinant of any matrix representation of A. This is a well-defined notion (i.e. independent of a choice of basis) since the determinant is invariant under similarity transformations.
As one might expect, it is possible to define the determinant of a linear transformation in a coordinate-free manner. If V is an n-dimensional vector space, then one can construct its top exterior power ΛnV. This is a one-dimensional vector space whose elements are written
where each vi is a vector in V and the wedge product ∧ is antisymmetric (i.e., u∧u = 0). Any linear transformation A : V → V induces a linear transformation of ΛnV as follows:
Since ΛnV is one-dimensional this operation is just multiplication by some scalar that depends on A. This scalar is called the determinant of A. That is, we define det(A) by the equation
One can check that this definition agrees with the coordinate-dependent definition given above.
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, determinants were first used in Chinese mathematics textbook The Nine Chapters on the Mathematical Art (九章算術, Chinese scholars, around the 3rd century BC). In Europe, two-by-two determinants were considered by Cardano at the end of the 16th century and larger ones by Leibniz and, in Japan, by Seki about 100 years later.[5][6]
In Japan, determinants were introduced to study elimination of variables in systems of higher-order algebraic equations. They used it to give short-hand representation for the resultant. After the first work by Seki in 1683, Laplace's formula was given by two independent groups of scholars: Tanaka, Iseki (算法発揮, Sampo-Hakki, published in 1690) and Seki, Takebe, Takebe (大成算経, taisei-sankei, written at least before 1710). However, doubts have been raised about how much they recognized the determinant as an independent object.
In Europe, Cramer (1750) added to the theory, treating the subject in relation to sets of equations. The recurrent law was first announced by Bézout (1764).
It was Vandermonde (1771) who first recognized determinants as independent functions.[5] Laplace (1772) [7][8] 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. Lagrange was the first to apply determinants 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 determinants (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.
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 m = 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. (See Cauchy-Binet formula.) In this he used the word determinant in its present sense[9][10], 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.[5][11] With him begins the theory in its generality.
The next important figure was Jacobi[6] (from 1827). He early used the functional determinant which Sylvester later called the Jacobian, and in his memoirs in Crelle 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.[12][13]
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 text-books on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.