In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix is commonly denoted,, or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism.
The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries.
The determinant of a matrix is
\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc,
\begin{vmatrix}a&b&c\ d&e&f\ g&h&i\end{vmatrix}=aei+bfg+cdh-ceg-bdi-afh.
The determinant of an matrix can be defined in several equivalent ways, the most common being Leibniz formula, which expresses the determinant as a sum of
n!
Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the matrices that has the four following properties:
The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns.
The determinant is invariant under matrix similarity. This implies that, given a linear endomorphism of a finite-dimensional vector space, the determinant of the matrix that represents it on a basis does not depend on the chosen basis. This allows defining the determinant of a linear endomorphism, which does not depends on the choice of a coordinate system.
Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry, the signed -dimensional volume of a -dimensional parallelepiped is expressed by a determinant, and the determinant of a linear endomorphism determines how the orientation and the -dimensional volume are transformed under the endomorphism. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.
The determinant of a matrix
\begin{pmatrix}a&b\\c&d\end{pmatrix}
\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=\begin{vmatrix}a&b\\c&d\end{vmatrix}=ad-bc.
\det\begin{pmatrix}3&7\\1&-4\end{pmatrix}=\begin{vmatrix}3&7\ 1&{-4}\end{vmatrix}=(3 ⋅ (-4))-(7 ⋅ 1)=-19.
The determinant has several key properties that can be proved by direct evaluation of the definition for
2 x 2
\begin{pmatrix}1&0\ 0&1\end{pmatrix}
\begin{vmatrix}a&b\ a&b\end{vmatrix}=ab-ba=0.
\begin{vmatrix}a&b+b'\ c&d+d'\end{vmatrix}=a(d+d')-(b+b')c=\begin{vmatrix}a&b\ c&d\end{vmatrix}+\begin{vmatrix}a&b'\ c&d'\end{vmatrix}.
r
\begin{vmatrix}r ⋅ a&b\ r ⋅ c&d\end{vmatrix}=rad-brc=r(ad-bc)=r ⋅ \begin{vmatrix}a&b\\c&d\end{vmatrix}.
If the matrix entries are real numbers, the matrix can be used to represent two linear maps: one that maps the standard basis vectors to the rows of, and one that maps them to the columns of . In either case, the images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram defined by the rows of the above matrix is the one with vertices at,,, and, as shown in the accompanying diagram.
The absolute value of is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by . (The parallelogram formed by the columns of is in general a different parallelogram, but since the determinant is symmetric with respect to rows and columns, the area will be the same.)
The absolute value of the determinant together with the sign becomes the oriented area of the parallelogram. The oriented area is the same as the usual area, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the identity matrix).
To show that is the signed area, one may consider a matrix containing two vectors and representing the parallelogram's sides. The signed area can be expressed as for the angle θ between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the sine this already is the signed area, yet it may be expressed more conveniently using the cosine of the complementary angle to a perpendicular vector, e.g., so that becomes the signed area in question, which can be determined by the pattern of the scalar product to be equal to according to the following equations:
Signedarea= |\boldsymbol{u}||\boldsymbol{v}|\sin\theta=\left|\boldsymbol{u}\perp\right|\left|\boldsymbol{v}\right|\cos\theta'= \begin{pmatrix}-b\ a\end{pmatrix} ⋅ \begin{pmatrix}c\ d\end{pmatrix}=ad-bc.
Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by A. When the determinant is equal to one, the linear mapping defined by the matrix is equi-areal and orientation-preserving.
The object known as the bivector is related to these ideas. In 2D, it can be interpreted as an oriented plane segment formed by imagining two vectors each with origin, and coordinates and . The bivector magnitude (denoted by) is the signed area, which is also the determinant .[1]
If an real matrix A is written in terms of its column vectors
A=\left[\begin{array}{c|c|c|c}a1&a2& … &an\end{array}\right]
A\begin{pmatrix}1\ 0\ \vdots\\0\end{pmatrix}=a1, A\begin{pmatrix}0\ 1\ \vdots\\0\end{pmatrix}=a2, \ldots, A\begin{pmatrix}0\\0\ \vdots\\1\end{pmatrix}=an.
This means that
A
a1,a2,\ldots,an,
P=\left\{c1a1+ … +cnan\mid0\leqci\leq1 \foralli\right\}.
The determinant gives the signed n-dimensional volume of this parallelotope,
\det(A)=\pmvol(P),
Let A be a square matrix with n rows and n columns, so that it can be written as
A=\begin{bmatrix} a1,1&a1,2& … &a1,n\\ a2,1&a2,2& … &a2,n\\ \vdots&\vdots&\ddots&\vdots\\ an,1&an,2& … &an,n\end{bmatrix}.
The entries
a1,1
The determinant of A is denoted by det(A), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:
\begin{vmatrix} a1,1&a1,2& … &a1,n\\ a2,1&a2,2& … &a2,n\\ \vdots&\vdots&\ddots&\vdots\\ an,1&an,2& … &an,n\end{vmatrix}.
There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question.
See main article: Leibniz formula for determinants.
The Leibniz formula for the determinant of a matrix is the following:
\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix} =aei+bfg+cdh-ceg-bdi-afh.
Generalizing the above to higher dimensions, the determinant of an
n x n
\{1,2,...,n\}
\sigma
\sigma(1),\sigma(2),\ldots,\sigma(n)
Sn
sgn(\sigma)
\sigma
+1,
-1.
Given a matrix
A=\begin{bmatrix} a1,1\ldotsa1,n\\ \vdots \vdots\\ an,1\ldotsan,n\end{bmatrix},
\det(A)=\begin{vmatrix} a1,1\ldotsa1,n\\ \vdots \vdots\\ an,1\ldotsan,n\end{vmatrix}=
\sum | |
\sigma\inSn |
sgn(\sigma)a1,\sigma(1) … an,\sigma(n).
Using pi notation for the product, this can be shortened into
\det(A)=
\sum | |
\sigma\inSn |
\left(sgn(\sigma)
n | |
\prod | |
i=1 |
ai,\sigma(i)\right)
\varepsilon | |
i1,\ldots,in |
\{1,\ldots,n\}
\det(A)=
\sum | |
i1,i2,\ldots,in |
\varepsilon | |
i1 … in |
a | |
1,i1 |
…
a | |
n,in |
,
\{1,\ldots,n\}.
The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an
n x n
n
A= (a1,...,an ),
ai
\det\left(I\right)=1
I
A
aj=r ⋅ v+w
\begin{align}|A| &= |a1,...,aj-1,r ⋅ v+w,aj+1,...,an|\\ &=r ⋅ |a1,...,v,...an|+|a1,...,w,...,an| \end{align}
|a1,...,v,...,v,...,an|=0.
If the determinant is defined using the Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any
n x n
To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a standard basis vector. These determinants are either 0 (by property 9) or else ±1 (by properties 1 and 12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear.
These rules have several further consequences:
n x n
A
A
aij=0
i>j
i<j
\sigma
These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact, Gaussian elimination can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix
A
A=\begin{bmatrix} -2&-1&2\\ 2&1&4\\ -3&3&-1 \end{bmatrix}.
Matrix | B=\begin{bmatrix} -3&-1&2\\ 3&1&4\\ 0&3&-1 \end{bmatrix} | C=\begin{bmatrix} -3&5&2\\ 3&13&4\\ 0&0&-1 \end{bmatrix} | D=\begin{bmatrix} 5&-3&2\\ 13&3&4\\ 0&0&-1 \end{bmatrix} | E=\begin{bmatrix} 18&-3&2\\ 0&3&4\\ 0&0&-1 \end{bmatrix} | |||||||||||||||||
Obtained by | add the second column to the first | add 3 times the third column to the second | swap the first two columns | add
| |||||||||||||||||
Determinant | A | = | B | B | = | C | D | = - | C | E | = | D |
Combining these equalities gives
|A|=-|E|=-(18 ⋅ 3 ⋅ (-1))=54.
The determinant of the transpose of
A
\det\left(Asf{T}\right)=\det(A)
The determinant is a multiplicative map, i.e., for square matrices
A
B
\det(AB)=\det(A)\det(B)
This key fact can be proven by observing that, for a fixed matrix
B
A
\detB
A
A matrix
A
\det\left(A-1\right)=
1 | |
\det(A) |
=[\det(A)]-1
In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size
n
K
\operatorname{GL}n(K)
\operatorname{SL}n(K)\subset\operatorname{GL}n(K)
Because the determinant respects multiplication and inverses, it is in fact a group homomorphism from
\operatorname{GL}n(K)
K x
K
\operatorname{SL}n(K)
\operatorname{SL}n(K)
\operatorname{GL}n(K)
\operatorname{GL}n(K)/\operatorname{SL}n(K)
K x
The Cauchy–Binet formula is a generalization of that product formula for rectangular matrices. This formula can also be recast as a multiplicative formula for compound matrices whose entries are the determinants of all quadratic submatrices of a given matrix.
Laplace expansion expresses the determinant of a matrix
A
Mi,j
(n-1) x (n-1)
A
i
j
(-1)i+jMi,j
i
\det(A)=
n | |
\sum | |
j=1 |
(-1)i+jai,jMi,j,
i=1
\begin{vmatrix}a&b&c\ d&e&f\ g&h&i\end{vmatrix}= a\begin{vmatrix}e&f\ h&i\end{vmatrix}-b\begin{vmatrix}d&f\ g&i\end{vmatrix}+c\begin{vmatrix}d&e\ g&h\end{vmatrix}
2 x 2
j
\det(A)=
n | |
\sum | |
i=1 |
(-1)i+jai,jMi,j.
\tbinomnk
\operatorname{adj}(A)
(\operatorname{adj}(A))i,j=(-1)i+jMji.
For every matrix, one has[6]
(\detA)I=A\operatorname{adj}A=(\operatorname{adj}A)A.
Thus the adjugate matrix can be used for expressing the inverse of a nonsingular matrix:
A-1=
1{\det | |
A}\operatorname{adj}A. |
The formula for the determinant of a
2 x 2
A,B,C,D
m x m
m x n
n x m
n x n
\det\begin{pmatrix}A&0\ C&D\end{pmatrix}=\det(A)\det(D)=\det\begin{pmatrix}A&B\ 0&D\end{pmatrix}.
A
\begin{align} \det\begin{pmatrix}A&B\ C&D\end{pmatrix} &=\det(A)\det\begin{pmatrix}A&B\ C&D\end{pmatrix} \underbrace{\det\begin{pmatrix}A-1&-A-1B\ 0&In\end{pmatrix}}
=\det(A-1)=(\detA)-1 |
\\ &=\det(A)\det\begin{pmatrix}Im&0\ CA-1&D-CA-1B\end{pmatrix}\\ &=\det(A)\det(D-CA-1B), \end{align}
\det(A)(D-CA-1B)
D
1 x 1
A similar result holds when
D
\begin{align} \det\begin{pmatrix}A&B\ C&D\end{pmatrix} &=\det(D)\det\begin{pmatrix}A&B\ C&D\end{pmatrix} \underbrace{\det\begin{pmatrix}Im&0\ -D-1C&D-1
\end{pmatrix}} | |
=\det(D-1)=(\detD)-1 |
\\ &=\det(D)\det\begin{pmatrix}A-BD-1C&BD-1\ 0&In\end{pmatrix}\\ &=\det(D)\det(A-BD-1C). \end{align}
If the blocks are square matrices of the same size further formulas hold. For example, if
C
D
CD=DC
\det\begin{pmatrix}A&B\ C&D\end{pmatrix}=\det(AD-BC).
2 x 2
For
A=D
B=C
A
B
\det\begin{pmatrix}A&B\ B&A\end{pmatrix}=\det(A-B)\det(A+B).
Sylvester's determinant theorem states that for A, an matrix, and B, an matrix (so that A and B have dimensions allowing them to be multiplied in either order forming a square matrix):
\det\left(Im+AB\right)=\det\left(In+BA\right),
where Im and In are the and identity matrices, respectively.
From this general result several consequences follow.
The determinant of the sum
A+B
A
B
C
Brunn–Minkowski theorem implies that the th root of determinant is a concave function, when restricted to Hermitian positive-definite
n x n
n x n
For the special case of
2 x 2
\det(A+B)=\det(A)+\det(B)+tr(A)tr(B)-tr(AB).
This has an application to
2 x 2
a
b
tr(i)=0
A=aI
B=bi
\det(aI+bi)=a2\det(I)+b2\det(i)=a2+b2.
tr(i)=0
\det(I)=\det(i)=1
The determinant is closely related to two other central concepts in linear algebra, the eigenvalues and the characteristic polynomial of a matrix. Let
A
n x n
A
λ1,λ2,\ldots,λn
\det(A)=
n | |
\prod | |
i=1 |
λi=λ1λ2 … λn.
From this, one immediately sees that the determinant of a matrix
A
0
A
A
0
A
The characteristic polynomial is defined as[12]
\chiA(t)=\det(t ⋅ I-A).
t
I
A
A
λ
\chiA(λ)=0.
A Hermitian matrix is positive definite if all its eigenvalues are positive. Sylvester's criterion asserts that this is equivalent to the determinants of the submatrices
Ak:=\begin{bmatrix} a1,1&a1,2& … &a1,k\\ a2,1&a2,2& … &a2,k\\ \vdots&\vdots&\ddots&\vdots\\ ak,1&ak,2& … &ak,k\end{bmatrix}
being positive, for all
k
1
n
The trace tr(A) is by definition the sum of the diagonal entries of and also equals the sum of the eigenvalues. Thus, for complex matrices,
\det(\exp(A))=\exp(\operatorname{tr}(A))
or, for real matrices,
\operatorname{tr}(A)=log(\det(\exp(A))).
Here exp denotes the matrix exponential of, because every eigenvalue of corresponds to the eigenvalue exp of exp. In particular, given any logarithm of, that is, any matrix satisfying
\exp(L)=A
the determinant of is given by
\det(A)=\exp(\operatorname{tr}(L)).
For example, for,, and, respectively,
\begin{align} \det(A)&=
1 | |
2 |
\left(\left(\operatorname{tr}(A)\right)2-\operatorname{tr}\left(A2\right)\right),\\ \det(A)&=
1 | |
6 |
\left(\left(\operatorname{tr}(A)\right)3-3\operatorname{tr}(A)~\operatorname{tr}\left(A2\right)+2\operatorname{tr}\left(A3\right)\right),\\ \det(A)&=
1 | |
24 |
\left(\left(\operatorname{tr}(A)\right)4-6\operatorname{tr}\left(A2\right)\left(\operatorname{tr}(A)\right)2+3\left(\operatorname{tr}\left(A2\right)\right)2+8\operatorname{tr}\left(A3\right)~\operatorname{tr}(A)-6\operatorname{tr}\left(A4\right)\right). \end{align}
cf. Cayley-Hamilton theorem. Such expressions are deducible from combinatorial arguments, Newton's identities, or the Faddeev–LeVerrier algorithm. That is, for generic, the signed constant term of the characteristic polynomial, determined recursively from
cn=1;~~~cn-m=-
1 | |
m |
m | |
\sum | |
k=1 |
cn-m+k\operatorname{tr}\left(Ak\right)~~(1\lem\len)~.
In the general case, this may also be obtained from[13]
\det(A)=\sum\begin{array{c}k1,k2,\ldots,kn\geq0\\k1+2k2+ … +nkn=n\end{array}}\prod
n | |
l=1 |
| ||||||
|
\operatorname{tr}\left(Al\right)
kl | |
,
where the sum is taken over the set of all integers satisfying the equation
n | |
\sum | |
l=1 |
lkl=n.
The formula can be expressed in terms of the complete exponential Bell polynomial of n arguments sl = −(l – 1)! tr(Al) as
\det(A)=
(-1)n | |
n! |
Bn(s1,s2,\ldots,sn).
This formula can also be used to find the determinant of a matrix with multidimensional indices and . The product and trace of such matrices are defined in a natural way as
I | |
(AB) | |
J |
=\sumK
I | |
A | |
K |
K | |
B | |
J, |
\operatorname{tr}(A)=\sumI
I | |
A | |
I. |
An important arbitrary dimension identity can be obtained from the Mercator series expansion of the logarithm when the expansion converges. If every eigenvalue of A is less than 1 in absolute value,
\det(I+A)=
infty | |
\sum | |
k=0 |
1 | |
k! |
infty | |
\left(-\sum | |
j=1 |
(-1)j | |
j |
\operatorname{tr}\left(Aj\right)\right)k,
where is the identity matrix. More generally, if
infty | |
\sum | |
k=0 |
1 | |
k! |
infty | |
\left(-\sum | |
j=1 |
(-1)jsj | |
j |
\operatorname{tr}\left(Aj\right)\right)k,
is expanded as a formal power series in then all coefficients of for are zero and the remaining polynomial is .
For a positive definite matrix, the trace operator gives the following tight lower and upper bounds on the log determinant
\operatorname{tr}\left(I-A-1\right)\lelog\det(A)\le\operatorname{tr}(A-I)
with equality if and only if . This relationship can be derived via the formula for the Kullback-Leibler divergence between two multivariate normal distributions.
Also,
n | |
\operatorname{tr |
\left(A-1\right)}\leq
| ||||
\det(A) |
\leq
1 | |
n |
\operatorname{tr}(A)\leq\sqrt{
1 | |
n |
\operatorname{tr}\left(A2\right)}.
These inequalities can be proved by expressing the traces and the determinant in terms of the eigenvalues. As such, they represent the well-known fact that the harmonic mean is less than the geometric mean, which is less than the arithmetic mean, which is, in turn, less than the root mean square.
The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a polynomial function from
Rn
R
d\det(A) | |
d\alpha |
=\operatorname{tr}\left(\operatorname{adj}(A)
dA | |
d\alpha |
\right).
where
\operatorname{adj}(A)
A
A
d\det(A) | |
d\alpha |
=\det(A)\operatorname{tr}\left(A-1
dA | |
d\alpha |
\right).
Expressed in terms of the entries of
A
\partial\det(A) | |
\partialAij |
=\operatorname{adj}(A)ji=\det(A)\left(A-1\right)ji.
Yet another equivalent formulation is
\det(A+\epsilonX)-\det(A)=\operatorname{tr}(\operatorname{adj}(A)X)\epsilon+O\left(\epsilon2\right)=\det(A)\operatorname{tr}\left(A-1X\right)\epsilon+O\left(\epsilon2\right)
using big O notation. The special case where
A=I
\det(I+\epsilonX)=1+\operatorname{tr}(X)\epsilon+O\left(\epsilon2\right).
This identity is used in describing Lie algebras associated to certain matrix Lie groups. For example, the special linear group
\operatorname{SL}n
\detA=1
ak{sl}n
Writing a
3 x 3
A=\begin{bmatrix}a&b&c\end{bmatrix}
a,b,c
\begin{align} \nablaa\det(A)&=b x c\\ \nablab\det(A)&=c x a\\ \nablac\det(A)&=a x b. \end{align}
Historically, determinants were used long before matrices: A determinant was originally 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 Chinese mathematics textbook The Nine Chapters on the Mathematical Art (九章算術, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed by Cardano in 1545 by a determinant-like entity.
Determinants proper originated separately from the work of Seki Takakazu in 1683 in Japan and parallelly of Leibniz in 1693.[14] [15] stated, without proof, Cramer's rule. Both Cramer and also were led to determinants by the question of plane curves passing through a given set of points.
Vandermonde (1771) first recognized determinants as independent functions.[16] gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case.[17] Immediately following, Lagrange (1773) treated determinants of the second and third order and applied it 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 "determinant" (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 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,[18] [19] 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.[20] With him begins the theory in its generality.
used the functional determinant which Sylvester later called the Jacobian. In his memoirs in Crelle's Journal 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. introduced the modern notation for the determinant using vertical bars.[21]
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 textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.
Determinants can be used to describe the solutions of a linear system of equations, written in matrix form as
Ax=b
x
\det(A)
xi=
\det(Ai) | |
\det(A) |
i=1,2,3,\ldots,n
where
Ai
i
A
b
\det(Ai)= \det\begin{bmatrix}a1&\ldots&b&\ldots&an\end{bmatrix}=
n | |
\sum | |
j=1 |
xj\det\begin{bmatrix}a1&\ldots&ai-1&aj&ai+1&\ldots&an\end{bmatrix}= xi\det(A)
aj
A\operatorname{adj}(A)=\operatorname{adj}(A)A=\det(A)In.
Cramer's rule can be implemented in
\operatornameO(n3)
Determinants can be used to characterize linearly dependent vectors:
\detA
A
v1,v2\inR3
v3
3 x 3
f1(x),...,fn(x)
n-1
W(f1,\ldots,fn)(x)= \begin{vmatrix} f1(x)&f2(x)& … &fn(x)\\ f1'(x)&f2'(x)& … &fn'(x)\\ \vdots&\vdots&\ddots&\vdots\\
(n-1) | |
f | |
1 |
(x)&
(n-1) | |
f | |
2 |
(x)& … &
(n-1) | |
f | |
n |
(x) \end{vmatrix}.
It is non-zero (for some
x
n-1
See main article: Orientation (vector space). The determinant can be thought of as assigning a number to every sequence of n vectors in Rn, by using the square matrix whose columns are the given vectors. The determinant will be nonzero if and only if the sequence of vectors is a basis for Rn. In that case, the sign of the determinant determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis. In the case of an orthogonal basis, the magnitude of the determinant is equal to the product of the lengths of the basis vectors. For instance, an orthogonal matrix with entries in Rn represents an orthonormal basis in Euclidean space, and hence has determinant of ±1 (since all the vectors have length 1). The determinant is +1 if and only if the basis has the same orientation. It is −1 if and only if the basis has the opposite orientation.
More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal or matrix, this is a rotation), while if it is negative, A switches the orientation of the basis.
As pointed out above, 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
f:Rn\toRn
A
S\subsetRn
f(S)
|\det(A)|
S
f:Rn\toRm
m x n
A
n
f(S)
\operatorname{volume}(f(S))=\sqrt{\det\left(Asf{T}A\right)}\operatorname{volume}(S).
a,b,c,d
1 | |
6 |
⋅ |\det(a-b,b-c,c-d)|
For a general differentiable function, much of the above carries over by considering the Jacobian matrix of f. For
f:Rn → Rn,
the Jacobian matrix is the matrix whose entries are given by the partial derivatives
D(f)=\left(
\partialfi | |
\partialxj |
\right)1.
Its determinant, the Jacobian determinant, appears in the higher-dimensional version of integration by substitution: for suitable functions f and an open subset U of Rn (the domain of f), the integral over f(U) of some other function is given by
\intf(U)\phi(v)dv=\intU\phi(f(u))\left|\det(\operatorname{D}f)(u)\right|du.
The Jacobian also occurs in the inverse function theorem.
When applied to the field of Cartography, the determinant can be used to measure the rate of expansion of a map near the poles.[22]
The above identities concerning the determinant of products and inverses of matrices imply that similar matrices have the same determinant: two matrices A and B are similar, if there exists an invertible matrix X such that . Indeed, repeatedly applying the above identities yields
\det(A)=\det(X)-1\det(B)\det(X)=\det(B)\det(X)-1\det(X)=\det(B).
The determinant is therefore also called a similarity invariant. The determinant of a linear transformation
T:V\toV
R
Z
\det(I)=1
A matrix
A\in\operatorname{Mat}n(R)
R
R
R=Z
The determinant being multiplicative, it defines a group homomorphism
\operatorname{GL}n(R) → R x ,
n x n
R
R
f:R\toS
\operatorname{GL}n(f):\operatorname{GL}n(R)\to\operatorname{GL}n(S)
R
f
f(\det((ai,j)))=\det((f(ai,j)))
For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo
m
m
\operatorname{GL}n
(-) x
\det:\operatorname{GL}n\toGm.
The determinant of a linear transformation
T:V\toV
n
V
n
R
n
wedgenV
V
T
\begin{align} wedgenT:wedgenV& → wedgenV\\ v1\wedgev2\wedge...\wedgevn&\mapstoTv1\wedgeTv2\wedge...\wedgeTvn. \end{align}
As
wedgenV
wedgenT
R
R
vi\inV
\left(wedgenT\right)\left(v1\wedge...\wedgevn\right)=\det(T) ⋅ v1\wedge...\wedgevn.
This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on
n
Rn
wedgenV
V
wedgekV
k<n
Determinants as treated above admit several variants: the permanent of a matrix is defined as the determinant, except that the factors
sgn(\sigma)
Sn
A
F
\det:A\toF.
A=\operatorname{Mat}n(F)
\det(a+ib+jc+kd)=a2+b2+c2+d2
NL/F:L\toF
For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated. Functional analysis provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators.
The Fredholm determinant defines the determinant for operators known as trace class operators by an appropriate generalization of the formula
\det(I+A)=\exp(\operatorname{tr}(log(I+A))).
Another infinite-dimensional notion of determinant is the functional determinant.
For operators in a finite factor, one may define a positive real-valued determinant called the Fuglede−Kadison determinant using the canonical trace. In fact, corresponding to every tracial state on a von Neumann algebra there is a notion of Fuglede−Kadison determinant.
For matrices over non-commutative rings, multilinearity and alternating properties are incompatible for,[25] so there is no good definition of the determinant in this setting.
For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero with a regular element of R as value on some pair of arguments implies that R is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably quasideterminants and the Dieudonné determinant. For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include the q-determinant on quantum groups, the Capelli determinant on Capelli matrices, and the Berezinian on supermatrices (i.e., matrices whose entries are elements of
Z2
Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for applications such as checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques.[26] Computational geometry, however, does frequently use calculations related to determinants.
While the determinant can be computed directly using the Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating
n!
n
n x n
n!
Gaussian elimination consists of left multiplying a matrix by elementary matrices for getting a matrix in a row echelon form. One can restrict the computation to elementary matrices of determinant . In this case, the determinant of the resulting row echelon form equals the determinant of the initial matrix. As a row echelon form is a triangular matrix, its determinant is the product of the entries of its diagonal.
So, the determinant can be computed for almost free from the result of a Gaussian elemination.
Some methods compute
\det(A)
\operatornameO(n3)
\operatornameO(n!)
For example, LU decomposition expresses
A
A=PLU.
P
1
L
U
L
U
P
\varepsilon
+1
-1
A
\det(A)=\varepsilon\det(L) ⋅ \det(U).
The order
\operatornameO(n3)
n
M(n)
M(n)\gena
a>2
O(M(n))
\operatornameO(n2.376)
In addition to the complexity of the algorithm, further criteria can be used to compare algorithms.Especially for applications concerning matrices over rings, algorithms that compute the determinant without any divisions exist. (By contrast, Gauss elimination requires divisions.) One such algorithm, having complexity
\operatornameO(n4)
\operatornameO(n3)
n
If the determinant of A and the inverse of A have already been computed, the matrix determinant lemma allows rapid calculation of the determinant of, where u and v are column vectors.
Charles Dodgson (i.e. Lewis Carroll of Alice's Adventures in Wonderland fame) invented a method for computing determinants called Dodgson condensation. Unfortunately this interesting method does not always work in its original form.[30]
ab= ab\begin{vmatrix}1&0\ 0&1\end{vmatrix}= a\begin{vmatrix}1&0\ 0&b\end{vmatrix}= \begin{vmatrix}a&0\ 0&b\end{vmatrix}= b\begin{vmatrix}a&0\ 0&1\end{vmatrix}= ba\begin{vmatrix}1&0\ 0&1\end{vmatrix}=ba,
a contradiction. There is no useful notion of multi-linear functions over a non-commutative ring.