Mathematics formula
In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements. If
is an
matrix, where
is the entry in the
-th row and
-th column of
, the formula is
![{\displaystyle \det(A)=\sum _{\tau \in S_{n}}\operatorname {sgn}(\tau )\prod _{i=1}^{n}a_{i\tau (i)}=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{\sigma (i)i}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/3a17a5e0dbdba67fb0555848d7388c4621778c5b)
where
is the sign function of permutations in the permutation group
, which returns
and
for even and odd permutations, respectively.
Another common notation used for the formula is in terms of the Levi-Civita symbol and makes use of the Einstein summation notation, where it becomes
![{\displaystyle \det(A)=\epsilon _{i_{1}\cdots i_{n}}{a}_{1i_{1}}\cdots {a}_{ni_{n}},}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/dfcc5e78b81f915398d7fb5819fca2927a2bb484)
which may be more familiar to physicists.
Directly evaluating the Leibniz formula from the definition requires
operations in general—that is, a number of operations asymptotically proportional to
factorial—because
is the number of order-
permutations. This is impractically difficult for even relatively small
. Instead, the determinant can be evaluated in
operations by forming the LU decomposition
(typically via Gaussian elimination or similar methods), in which case
and the determinants of the triangular matrices
and
are simply the products of their diagonal entries. (In practical applications of numerical linear algebra, however, explicit computation of the determinant is rarely required.) See, for example, Trefethen & Bau (1997). The determinant can also be evaluated in fewer than
operations by reducing the problem to matrix multiplication, but most such algorithms are not practical.
Theorem.
There exists exactly one function
which is alternating multilinear w.r.t. columns and such that
.
Proof.
Uniqueness: Let
be such a function, and let
be an
matrix. Call
the
-th column of
, i.e.
, so that
Also, let
denote the
-th column vector of the identity matrix.
Now one writes each of the
's in terms of the
, i.e.
.
As
is multilinear, one has
![{\displaystyle {\begin{aligned}F(A)&=F\left(\sum _{k_{1}=1}^{n}a_{k_{1}}^{1}E^{k_{1}},\dots ,\sum _{k_{n}=1}^{n}a_{k_{n}}^{n}E^{k_{n}}\right)=\sum _{k_{1},\dots ,k_{n}=1}^{n}\left(\prod _{i=1}^{n}a_{k_{i}}^{i}\right)F\left(E^{k_{1}},\dots ,E^{k_{n}}\right).\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7e734abe8031042835991d33ab22474544af92bb)
From alternation it follows that any term with repeated indices is zero. The sum can therefore be restricted to tuples with non-repeating indices, i.e. permutations:
![{\displaystyle F(A)=\sum _{\sigma \in S_{n}}\left(\prod _{i=1}^{n}a_{\sigma (i)}^{i}\right)F(E^{\sigma (1)},\dots ,E^{\sigma (n)}).}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a23bad4f447f6769e59e9d73e96981378a2d16ab)
Because F is alternating, the columns
can be swapped until it becomes the identity. The sign function
is defined to count the number of swaps necessary and account for the resulting sign change. One finally gets:
![{\displaystyle {\begin{aligned}F(A)&=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\left(\prod _{i=1}^{n}a_{\sigma (i)}^{i}\right)F(I)\\&=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{\sigma (i)}^{i}\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/79a84da98e304d02048847954e1137f47a2d5fad)
as
is required to be equal to
.
Therefore no function besides the function defined by the Leibniz Formula can be a multilinear alternating function with
.
Existence: We now show that F, where F is the function defined by the Leibniz formula, has these three properties.
Multilinear:
![{\displaystyle {\begin{aligned}F(A^{1},\dots ,cA^{j},\dots )&=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )ca_{\sigma (j)}^{j}\prod _{i=1,i\neq j}^{n}a_{\sigma (i)}^{i}\\&=c\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )a_{\sigma (j)}^{j}\prod _{i=1,i\neq j}^{n}a_{\sigma (i)}^{i}\\&=cF(A^{1},\dots ,A^{j},\dots )\\\\F(A^{1},\dots ,b+A^{j},\dots )&=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\left(b_{\sigma (j)}+a_{\sigma (j)}^{j}\right)\prod _{i=1,i\neq j}^{n}a_{\sigma (i)}^{i}\\&=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\left(\left(b_{\sigma (j)}\prod _{i=1,i\neq j}^{n}a_{\sigma (i)}^{i}\right)+\left(a_{\sigma (j)}^{j}\prod _{i=1,i\neq j}^{n}a_{\sigma (i)}^{i}\right)\right)\\&=\left(\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )b_{\sigma (j)}\prod _{i=1,i\neq j}^{n}a_{\sigma (i)}^{i}\right)+\left(\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{\sigma (i)}^{i}\right)\\&=F(A^{1},\dots ,b,\dots )+F(A^{1},\dots ,A^{j},\dots )\\\\\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8f8500375a1aff537d0445d180e43f0428cf68af)
Alternating:
![{\displaystyle {\begin{aligned}F(\dots ,A^{j_{1}},\dots ,A^{j_{2}},\dots )&=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\left(\prod _{i=1,i\neq j_{1},i\neq j_{2}}^{n}a_{\sigma (i)}^{i}\right)a_{\sigma (j_{1})}^{j_{1}}a_{\sigma (j_{2})}^{j_{2}}\\\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0eb7e9a8162b7cf5131c0b7bc5dc56aa85611ff7)
For any
let
be the tuple equal to
with the
and
indices switched.
![{\displaystyle {\begin{aligned}F(A)&=\sum _{\sigma \in S_{n},\sigma (j_{1})<\sigma (j_{2})}\left[\operatorname {sgn}(\sigma )\left(\prod _{i=1,i\neq j_{1},i\neq j_{2}}^{n}a_{\sigma (i)}^{i}\right)a_{\sigma (j_{1})}^{j_{1}}a_{\sigma (j_{2})}^{j_{2}}+\operatorname {sgn}(\sigma ')\left(\prod _{i=1,i\neq j_{1},i\neq j_{2}}^{n}a_{\sigma '(i)}^{i}\right)a_{\sigma '(j_{1})}^{j_{1}}a_{\sigma '(j_{2})}^{j_{2}}\right]\\&=\sum _{\sigma \in S_{n},\sigma (j_{1})<\sigma (j_{2})}\left[\operatorname {sgn}(\sigma )\left(\prod _{i=1,i\neq j_{1},i\neq j_{2}}^{n}a_{\sigma (i)}^{i}\right)a_{\sigma (j_{1})}^{j_{1}}a_{\sigma (j_{2})}^{j_{2}}-\operatorname {sgn}(\sigma )\left(\prod _{i=1,i\neq j_{1},i\neq j_{2}}^{n}a_{\sigma (i)}^{i}\right)a_{\sigma (j_{2})}^{j_{1}}a_{\sigma (j_{1})}^{j_{2}}\right]\\&=\sum _{\sigma \in S_{n},\sigma (j_{1})<\sigma (j_{2})}\operatorname {sgn}(\sigma )\left(\prod _{i=1,i\neq j_{1},i\neq j_{2}}^{n}a_{\sigma (i)}^{i}\right)\underbrace {\left(a_{\sigma (j_{1})}^{j_{1}}a_{\sigma (j_{2})}^{j_{2}}-a_{\sigma (j_{1})}^{j_{2}}a_{\sigma (j_{2})}^{j_{_{1}}}\right)} _{=0{\text{, if }}A^{j_{1}}=A^{j_{2}}}\\\\\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/3355a381e8cbfa6bcb10150c9ee4fa675a474183)
Thus if
then
.
Finally,
:
![{\displaystyle {\begin{aligned}F(I)&=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\prod _{i=1}^{n}I_{\sigma (i)}^{i}=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\prod _{i=1}^{n}\operatorname {\delta } _{i,\sigma (i)}\\&=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )\operatorname {\delta } _{\sigma ,\operatorname {id} _{\{1\ldots n\}}}=\operatorname {sgn}(\operatorname {id} _{\{1\ldots n\}})=1\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/25c3289e9cc17c5fd0ea82a6acbb8ad13a2eda3f)
Thus the only alternating multilinear functions with
are restricted to the function defined by the Leibniz formula, and it in fact also has these three properties. Hence the determinant can be defined as the only function
with these three properties.