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determinant

(Definition)

Overview

The determinant is an algebraic operation that transforms a square matrix into a scalar. This operation has many useful and important properties. For example, the determinant is zero if and only the matrix is singular (no inverse exists). The determinant also has an important geometric interpretation as the area of a parallelogram, and more generally as the volume of a higher-dimensional parallelepiped.

The notion of determinant predates matrices and linear transformations. Originally, the determinant was a number associated to a system of linear equations in variables. This number “determined” whether the system possessed a unique solution. In this sense, two-by-two determinants were considered by Cardano at the end of the 16th century and ones of arbitrary size (see the definition below) by Leibniz about 100 years later.

Definition

Let be an $n\times n$ matrix with entries $M_{ij}$ that are elements of a given field ¹. The determinant of , or $\det M$ for short, is the scalar quantity

$\displaystyle \det M = \left\vert \begin{array}{cccc} M_{11} & M_{12} &\ldots &... ...t = \sum_{\pi\in S_n} \mathrm{sgn}(\pi) M_{1\pi_1} M_{2\pi_2}\cdots M_{n\pi_n}.$

(1)

The index $\pi$ in the above sum varies over all the permutations of $\{1,\ldots,n\}$ (i.e., the elements of the symmetric group

.) Hence, there are

terms in the defining sum of the determinant. The symbol $\operatorname{sgn}(\pi)$ denotes the parity of the permutation; it is $\pm 1$ according to whether $\pi$ is an even or odd permutation. Using the Einstein summation convention one can also express the above definition as

$\displaystyle \det M = \epsilon_{\pi_1\pi_2\dots \pi_n} M^{\pi_1}{}_1 M^{\pi_2}{}_2 \cdots M^{\pi_n}{}_{n},$

(2)

where we've raised the first index so that $M^i{}_j = M_{ij}$ , and where

$\displaystyle \epsilon_{\pi_1\dots \pi_n} = \operatorname{sgn}(\pi)$

is known as the Levi-Civita permutation symbol.

By way of example, the determinant of a $2\times2$ matrix is given by

$\displaystyle \left\vert \begin{matrix}M_{11} & M_{12} \\ M_{21}& M_{22} \end{matrix}\right\vert = M_{11} M_{22}-M_{12}M_{21},$

There are six permutations of the numbers

, namely

$\displaystyle 1\overset{+}23,\;2\overset{+}31,\;3\overset{+}12,\;1\overset{-}32,\; 3\overset{-}21,\;2\overset{-}13;$

the overset sign indicates the permutation's signature. Accordingly, the $3\times 3$ deterimant is a sum of the following

terms:

$\begin{displaymath}\left\vert \begin{matrix} M_{11} & M_{12} & M_{13} \ M_{21}... ...{23}M_{32}-M_{13}M_{22} M_{31} - M_{12}M_{21}M_{33} \end{array}\end{displaymath}$

Remarks and important properties

The determinant operation converts matrix multiplication into scalar multiplication;
$\displaystyle \det(AB)=\det(A) \det(B),$
where are square matrices of the same size.
The determinant operation is multi-linear, and anti-symmetric with respect to the matrix's rows and columns. See the multi-linearity attachment for more details.
The determinant of a lower triangular, or an upper triangular matrix is the product of the diagonal entries, since all the other summands in (1) are zero.
Similar matrices have the same determinant. To be more precise, let and be square matrices with invertible. Then,
$\displaystyle \det(XAX^{-1})= \det(A).$
In particular, if we let be the matrix representing a change of basis, this shows that the determinant is independent of the basis. The same is true of the trace of a matrix. In fact, the whole characteristic polynomial of an endomorphism is definable without using a basis or a matrix, and it turns out that the determinant and trace are two of its coefficients.
The determinant of a matrix is zero if and only if is singular; that is, if there exists a non-trivial solution to the homogeneous equation
$\displaystyle A\mathbf{x} = \mathbf{0}.$
The transpose operation does not change the determinant:
$\displaystyle \det A^{\scriptscriptstyle\mathrm{T}} = \det A.$
The determinant of a diagonalizable transformation is equal to the product of its eigenvalues, counted with multiplicities.
The determinant is homogeneous of degree . This means that
$\displaystyle \det(k M ) = k^n \det M,\quad k\;$ is a scalar.

Footnotes

...¹: Most scientific and geometric applications deal with matrices made up of real or complex numbers. However, the determinant of a matrix over any field is well defined sense and has all the properties of the more conventional determinant. Indeed, many properties of the determinant remain valid for matrices with entries in a commutative ring.

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"determinant" is owned by rmilson. [ full author list (3) | owner history (3) ]

(view preamble)

Attachments:: cofactor expansion (Theorem) by rmilson
determinant as a multilinear mapping (Theorem) by rmilson
determinants of some matrices of special form (Result) by bwebste
n'th derivative of a determinant (Result) by GeraW
determinant condition for a sequence of vectors (Result) by GeraW
determinant inequalities (Result) by Andrea Ambrosio
determinant in terms of traces of powers (Theorem) by Mathprof
rule of Sarrus (Result) by pahio
lecture notes on determinants (Topic) by rmilson

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Cross-references: homogeneous of degree, multiplicities, eigenvalues, transformation, diagonalizable, transpose, homogeneous equation, coefficients, trace, definable, endomorphism, characteristic polynomial, trace of a matrix, basis, independent, change of basis, invertible, diagonal, product, upper triangular matrix, lower triangular, columns, rows, anti-symmetric, multi-linear, multiplication, matrix multiplication, signature, Levi-Civita permutation symbol, Einstein summation convention, odd permutation, even, parity, terms, symmetric group, permutations, sum, index, commutative ring, well defined, complex numbers, real, field, size, solution, variables, linear equations, linear transformations, parallelepiped, volume, parallelogram, area, interpretation, inverse, singular, matrix, properties, scalar, square matrix, Transforms, operation, algebraic
There are 122 references to this entry.

This is version 21 of determinant, born on 2002-03-20, modified 2006-01-12.
Object id is 2797, canonical name is Determinant2.
Accessed 45321 times total.

Classification:

AMS MSC:

15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions)

Pending Errata and Addenda

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