In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, the Binet–Cauchy identity, named after
Jacques Philippe Marie Binet
Jacques Philippe Marie Binet (; 2 February 1786 – 12 May 1856) was a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856. He made significant contributions to number theory, and the mathematical founda ...
and
Augustin-Louis Cauchy, states that
for every choice of
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s (or more generally, elements of a
commutative ring).
Setting and , it gives
Lagrange's identity
In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is:
\begin
\left( \sum_^n a_k^2\right) \left(\sum_^n b_k^2\right) - \left(\sum_^n a_k b_k\right)^2 & = \sum_^ \sum_^n \left(a_i b_j - a_j b_i\right)^2 \\
& \left(= \frac \sum_^n ...
, which is a stronger version of the
Cauchy–Schwarz inequality for the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. The Binet-Cauchy identity is a special case of the
Cauchy–Binet formula In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so tha ...
for matrix determinants.
The Binet–Cauchy identity and exterior algebra
When , the first and second terms on the right hand side become the squared magnitudes of
dot and
cross products respectively; in dimensions these become the magnitudes of the dot and
wedge product
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by convert ...
s. We may write it
where , , , and are vectors. It may also be written as a formula giving the dot product of two wedge products, as
which can be written as
in the case.
In the special case and , the formula yields
When both and are unit vectors, we obtain the usual relation
where is the angle between the vectors.
This is a special case of the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on the exterior algebra of a vector space, which is defined on wedge-decomposable elements as the
Gram determinant of their components.
Einstein notation
A relationship between the
Levi–Cevita symbols and the generalized
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 & ...
is
The
form of the Binet–Cauchy identity can be written as
Proof
Expanding the last term,
where the second and fourth terms are the same and artificially added to complete the sums as follows:
This completes the proof after factoring out the terms indexed by ''i''.
Generalization
A general form, also known as the
Cauchy–Binet formula In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so tha ...
, states the following:
Suppose ''A'' is an ''m''×''n''
matrix and ''B'' is an ''n''×''m'' matrix. If ''S'' is a
subset of with ''m'' elements, we write ''A
S'' for the ''m''×''m'' matrix whose columns are those columns of ''A'' that have indices from ''S''. Similarly, we write ''B
S'' for the ''m''×''m'' matrix whose ''rows'' are those rows of ''B'' that have indices from ''S''.
Then the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
of the
matrix product
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
of ''A'' and ''B'' satisfies the identity
where the sum extends over all possible subsets ''S'' of with ''m'' elements.
We get the original identity as special case by setting
Notes
References
*
*
{{DEFAULTSORT:Binet-Cauchy Identity
Mathematical identities
Multilinear algebra
Articles containing proofs