# Dictionary Definition

determinant adj : having the power or quality of
deciding; "the crucial experiment"; "cast the deciding vote"; "the
determinative (or determinant) battle" [syn: crucial, deciding(a),
determinative,
determining(a)]

### Noun

1 a determining or causal element or factor;
"education is an important determinant of one's outlook on life"
[syn: determiner,
determinative,
determining
factor, causal
factor]

2 a square matrix used to solve simultaneous
equations

# User Contributed Dictionary

## English

### Noun

- A determining factor; an element that determines the nature of something
- The unique scalar function over square matrices which is distributive over matrix multiplication, multilinear in the rows and columns, and takes the value of 1 for the unit matrix. Abbreviation: det
- A substance that causes a cell to adopt a particular fate.

#### Translations

determining factor

- Finnish: ratkaiseva tekijä
- Hungarian: meghatározó

in mathematical sense

- Finnish: determinantti
- French: déterminant
- German: Determinante
- Hungarian: determináns
- Icelandic: ákveða
- Italian: determinante
- Japanese: (gyōretsu-shiki)
- Slovak: determinant
- Swedish: determinant

## Slovak

- determining factor
- determinant

## Swedish

# Extensive Definition

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
as the scale factor
for volume 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 positive 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.

## Vertical bar notation

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 A = \begin a & b & c\\d & e
& f\\g & h & i \end\,

the determinant \det(A) might be indicated by |A|
or more explicitly as |A| = \begin a & b & c\\d & e
& f\\g & h & i \end.\,

That is, the square braces around the matrices
are replaced with elongated vertical bars.

## Determinants of 2-by-2 matrices

The 2×2 matrix,

A = \begin a & b\\c & d \end\,

has determinant

- \det(A)=ad-bc.\,

The interpretation when the matrix has real
number entries is that this gives 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 x^T A, where x 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
\begin 0 & 0 \end, \begin 0 & 1 \end, \begin 1 & 0 \end
and \begin1 & 1\end in turn. These row vectors define the
vertices of the unit square. With the more common matrix-vector
product Ax, the parallelogram has vertices at \begin 0 \\ 0 \end,
\begin a \\ c \end, \begin a+b \\ c+d \end and \begin b \\ d \end
(note that Ax = (x^T A^T)^T).

A formula for larger matrices will be given
below.

## Determinants of 3-by-3 matrices

The 3×3 matrix:

- A=\begina&b&c\\

- \begin

which can be remembered 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:

\begin \colora & \colorb & \colorc &
a & b \\ d & \colore & \colorf & \colord & e \\
g & h & \colori & \colorg & \colorh \end \quad -
\quad \begin a & b & \colorc & \colora & \colorb \\
d & \colore & \colorf & \colord & e \\ \colorg
& \colorh & \colori & g & h \end

Note that this mnemonic does not carry over into
higher dimensions.

## Applications

Determinants are used to characterize invertible
matrices (i.e., exactly those matrices with non-zero
determinants), and to explicitly describe the solution to a system
of linear
equations with Cramer's
rule. They can be used to find the eigenvalues of the matrix A
through the characteristic
polynomial

- p(x) = \det(xI - A) \,

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 n vectors in
\Bbb^n, 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 f: \Bbb^n \rightarrow \Bbb^n is represented by the matrix
A, and S is any measurable
subset of \Bbb^n, then
the volume of f(S) is given by \left| \det(A) \right| \times
\operatorname(S). More generally, if the linear map f: \Bbb^n
\rightarrow \Bbb^m is represented by the m-by-n matrix A, and S is
any measurable subset of \Bbb^, then the n-dimensional volume of f(S) is
given by \sqrt \times \operatorname(S). 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.

## General definition and computation

The definition of the determinant comes from the
following Theorem.

Theorem. Let Mn(K) denote the set of all n \times
n matrices over the field K. There exists exactly one
function

- F : M_n(K) \longrightarrow K

with the two properties:

- F is alternating multilinear with regard to columns;
- F(I) = 1.

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:

- \det(A) = \sum_ \sgn(\sigma) \prod_^n A_.

Here the sum is computed over all permutations \sigma of the
numbers and \sgn(\sigma) denotes the signature
of the permutation \sigma: +1 if \sigma is an
even permutation and −1 if it is
odd. \sigma: 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 n! (factorial) summands, and it is
therefore impractical to use it to calculate determinants for large
n.

For small matrices, one obtains the following
formulas:

- if A is a 1-by-1 matrix, then \det(A) = A_. \,

- if A is a 2-by-2 matrix, then \det(A) = A_A_ - A_A_. \,

- for a 3-by-3 matrix A, the formula is more complicated:

\begin \det(A) & = & A_A_A_ + A_A_A_ +
A_A_A_\\ & & - A_A_A_ - A_A_A_ - A_A_A_. \end\, which takes
the shape of the Sarrus'
scheme.

In general, determinants can be computed using
Gaussian
elimination using the following rules:

- If A is a triangular matrix, i.e. A_ = 0 \, whenever i > j or, alternatively, whenever i , then \det(A) = A_ A_ \cdots A_ \, (the product of the diagonal entries of A).
- 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 to another row, or a multiple of one column to another 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, which is efficient for relatively small matrices. To
do this along row i, say, we write

- \det(A) = \sum_^n A_C_ = \sum_^n A_ (-1)^ M_

where the C_ represent the matrix cofactors,
i.e. C_ is (-1)^ times the minor M_,
which is the determinant of the matrix that results from A by
removing the i-th row and the j-th column.

## Example

Suppose we want to compute the determinant
of

- A = \begin-2&2&-3\\

We can go ahead and use the Leibniz formula
directly:

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:

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:

- \begin0&2&-3\\

and this determinant can be quickly expanded
along the first 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(rI_n) = r^n \, and
thus

- \det(rA) = \det(rI_n \cdot A) = r^n \det(A) \, for all n-by-n matrices A and all scalars r.

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

- \det(A^) = \det(A)^. \,

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:

- \det(A^\mathrm) = \det(A). \,

The determinants of a complex matrix and of its
conjugate
transpose are conjugate:

- \det(A^*) = \det(A)^*. \,

The determinant of a matrix A exhibits the
following properties under
elementary matrix transformations of A:

- Exchanging rows or columns multiplies the determinant by −1.
- Multiplying a row or column by m multiplies the determinant by m.
- Adding a multiple of a row or column to another leaves the determinant unchanged.

This follows from the multiplicative property and
the determinants of the
elementary matrix transformation matrices.

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

- \det(A) = \det(B). \,

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 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) = \lambda_\lambda_ \cdots \lambda_.\,

This follows from the fact that A is always
similar to its Jordan
normal form, an upper triangular matrix with the eigenvalues on
the main diagonal.

### Useful identities

Sylvester's determinant theorem states that for any m-by-n
matrices A and B,

- \left.\det(I_m + A B^T) = \det(I_n + B^T A)\right. .

For the case of (column) vectors a and b, this
equality becomes

- \left.\det(I + a b^T) = 1 + b^T a\right. .

With X a nonsingular m-by-m matrix, this last
expression generalizes to

- \det(X + a b^T) = \det(X)\ (1 + b^T X^ a) .

Proofs can be found in http://www.ee.ic.ac.uk/hp/staff/www/matrix/proof003.html.

### Block matrices

Suppose, A, B, C, D are n\times n, n\times m,
m\times n, m\times m matrices respectively. Then

- \det\beginA& 0\\ C& D\end = \det\beginA& B\\ 0& D\end = \det(A) \det(D) .

- \beginA& B\\ C& D\end = \beginA& 0\\ C& 1\end \begin1& A^ B\\ 0& D - C A^ B\end

- \det\beginA& B\\ C& D\end = \det(A) \det(D - C A^ B) .

If d_ are diagonal matrices, then

- \det\begind_ & \ldots & d_\\ \vdots & & \vdots\\ d_ & \ldots & d_ \end =

This is a special case of the theorem published
in http://www.mth.kcl.ac.uk/~jrs/gazette/blocks.pdf.

### Relationship to trace

From this 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(\operatorname(A)).

Performing the substitution \scriptstyle A
\,\mapsto\, \log A in the above equation yields

- \det(A) = \exp(\operatorname(\log A)), \

which is closely related to the Fredholm
determinant. Similarly,

- \operatorname(A) = \log(\det(\exp A)). \

For n-by-n matrices there are the
relationships:

- Case n = 1: \left.\det(A) = \operatorname(A)\right.

- Case n = 2: \left.

- Case n = 3: \left.

- Case n = 4: \left.

- \ldots

which are closely related to
Newton's identities.

### Derivative

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

- d \,\det(A) = \operatorname(\operatorname(A) \,dA)

where adj(A) denotes the adjugate of A. In particular,
if A is invertible, we have

- d \,\det(A) = \det(A) \,\operatorname(A^ \,dA).

In component form, these are

- \frac

When \epsilon is a small number these are
equivalent to

- \det(A + \epsilon X) - \det(A)

The special case where A is equal to the identity
matrix I yields

- \det(I + \epsilon X) = 1 + \operatorname(X) \epsilon +O(\epsilon^2).

A useful property in the case of 3 x 3 matrices
is the following:

A may be written as A = \begin\bar & \bar
& \bar\end where \bar, \bar, \bar are vectors, then the
gradient over one of the three vectors may be written as the cross
product of the other two:

- \nabla_\bar\det(A) = \bar \times \bar
- \nabla_\bar\det(A) = \bar \times \bar
- \nabla_\bar\det(A) = \bar \times \bar.

## Abstract formulation

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

- A:V\to V\,

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

- v_1 \wedge v_2 \wedge \cdots \wedge v_n

- v_1 \wedge v_2 \wedge \cdots \wedge v_n \mapsto Av_1 \wedge Av_2 \wedge \cdots \wedge Av_n.

- Av_1 \wedge Av_2 \wedge \cdots \wedge Av_n = (\det A)\,v_1 \wedge v_2 \wedge \cdots \wedge v_n.

## Algorithmic implementation

- The naive method of implementing an algorithm to compute the determinant is to use Laplace's formula for expansion by cofactors. This approach is extremely inefficient in general, however, as it is of order n! (n factorial) for an n×n matrix M.
- An improvement to order n3 can be achieved by using LU decomposition to write M = LU for triangular matrices L and U. Now, det M = det LU = det L det U, and since L and U are triangular the determinant of each is simply the product of its diagonal elements. Alternatively one can perform the Cholesky decomposition if possible or the QR decomposition and find the determinant in a similar fashion.
- 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 n4 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.
- What is not often discussed is the so-called "bit complexity" of the problem, i.e. how many bits of accuracy you need to store for intermediate values. For example, using Gaussian elimination, you can reduce the matrix to upper triangular form, then multiply the main diagonal to get the determinant (this is essentially a special case of the LU decomposition as above), but a quick calculation will show that the bit size of intermediate values could potentially become exponential. One could talk about when it is appropriate to round intermediate values, but an elegant way of calculating the determinant uses the Bareiss Algorithm, an exact-division method based on Sylvester's identity to give a run time of order n3 and bit complexity roughly the bit size of the original entries in the matrix times n.

## History

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 the 3rd century BC Chinese math textbook The Nine Chapters on the Mathematical Art. 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.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.
Laplace
(1772) 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, 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.

The next important figure was Jacobi

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.

## See also

## References

## External links

- MIT Linear Algebra Lecture on Determinants
- Linear Systems Chapter from "Fundamental Problems of Algorithmic Algebra" Chee Yap's chapter on Linear Systems describing implementation aspects of Determinant computation.
- Mahajan, Meena and V. Vinay, “Determinant: Combinatorics, Algorithms, and Complexity”, Chicago Journal of Theoretical Computer Science, v. 1997 article 5 (1997).
- Online Matrix Calculator Online Matrix calculator.
- Linear algebra: determinants. Compute determinants of matrices up to order 6 using Laplace expansion you choose.
- Free Determinant software

determinant in Arabic: محدد

determinant in Bulgarian: Детерминанта

determinant in Bengali: নির্ণায়ক

determinant in Catalan: Determinant
(matemàtiques)

determinant in Czech: Determinant

determinant in Danish: Determinant

determinant in German: Determinante
(Mathematik)

determinant in Estonian: Determinant

determinant in Spanish: Determinante
(matemática)

determinant in Esperanto: Determinanto

determinant in Persian: دترمینان

determinant in French: Déterminant
(mathématiques)

determinant in Korean: 행렬식

determinant in Icelandic: Ákveða

determinant in Italian: Determinante

determinant in Hebrew: דטרמיננטה

determinant in Lithuanian: Determinantas

determinant in Hungarian: Determináns

determinant in Dutch: Determinant

determinant in Japanese: 行列式

determinant in Polish: Wyznacznik

determinant in Portuguese: Determinante

determinant in Romanian: Determinant
(matematică)

determinant in Russian: Определитель

determinant in Serbian: Детерминанта

determinant in Swedish: Determinant

determinant in Thai: ดีเทอร์มิแนนต์

determinant in Vietnamese: Định thức

determinant in Ukrainian: Визначник

determinant in Urdu: دترمینان

determinant in Chinese: 行列式

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