Binet-Cauchy identity
Encyclopedia
In algebra
, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet
and Augustin-Louis Cauchy, states that
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, the Binet–Cauchy identity, named after Jacques Philippe Marie Binet
Jacques Philippe Marie Binet
Jacques Philippe Marie Binet 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 foundations of matrix algebra which would later lead to important contributions by Cayley...
and Augustin-Louis Cauchy, states that
-
for every choice of realReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
or complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s (or more generally, elements of a commutative ringCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
).
Setting ai = ci and bi = di, it gives the Lagrange's identity, which is a stronger version of the Cauchy–Schwarz inequalityCauchy–Schwarz inequalityIn mathematics, the Cauchy–Schwarz inequality , is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas...
for the Euclidean spaceEuclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
.
The Binet–Cauchy identity and exterior algebra
When n = 3 the first and second terms on the right hand side become the squared magnitudes of dotDot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
and cross productCross productIn mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
s respectively; in n dimensions these become the magnitudes of the dot and wedge products. We may write it
where a, b, c, and d are vectors. It may also be written as a formula giving the dot product of two wedge products, as
In the special case of unit vectors a=c and b=d, the formula yields
When both vectors are unit vectors, we obtain the usual relation
where φ is the angle between the vectors.
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, states the following:
Suppose A is an m×n matrixMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
and B is an n×m matrix. If S is a subsetSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of {1, ..., n} with m elements, we write AS for the m×m matrix whose columns are those columns of A that have indices from S. Similarly, we write BS for the m×m matrix whose rows are those rows of B that have indices from S.
Then the determinantDeterminantIn linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the matrix product of A and B satisfies the identity
where the sum extends over all possible subsets S of {1, ..., n} with m elements.
We get the original identity as special case by setting