Lagrange's identityFrom Wikipedia, the free encyclopedia

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In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is:[1][2]

which applies to any two sets {a1, a2, . . ., an} and {b1, b2, . . ., bn} of real or complex numbers (or more generally, elements of a commutative ring). This identity is a special

form of the Binet–Cauchy identity.

In a more compact vector notation, Lagrange's identity is expressed as:[3]

where a and b are n-dimensional vectors with components that are real numbers. The extension to complex numbers requires the interpretation of the dot product as an inner

product or Hermitian dot product. Explicitly, for complex numbers, Lagrange's identity can be written in the form:[4]

involving the absolute value.[5]

Since the right-hand side of the identity is clearly non-negative, it implies Cauchy's inequality in the finite-dimensional real coordinate space ℝn and its complex counterpart

ℂn.

Contents

1 Lagrange's identity and exterior algebra

2 Lagrange's identity and vector calculus

2.1 Seven dimensions

2.2 Quaternions

3 Proof of algebraic form

4 See also

5 References

Lagrange's identity and exterior algebra

In terms of the wedge product, Lagrange's identity can be written

Hence, it can be seen as a formula which gives the length of the wedge product of two vectors, which is the area of the paralleogram they define, in terms of the dot products

of the two vectors, as

Lagrange's identity and vector calculus

In three dimensions, Lagrange's identity asserts that the square of the area of a parallelogram in space is equal to the sum of the squares of its projections onto the Cartesian

coordinate planes. Algebraically, if a and b are vectors in ℝ3 with lengths |a| and |b|, then Lagrange's identity can be written in terms of the cross product and dot product:

[6][7]

Using the definition of angle based upon the dot product (see also Cauchy–Schwarz inequality), the left-hand side is

where θ is the angle formed by the vectors a and b. The area of a parallelogram with sides |a| and |b| and angle θ is known in elementary geometry to be

so the left-hand side of Lagrange's identity is the squared area of the parallelogram. The cross product appearing on the right-hand side is defined by

which is a vector whose components are equal in magnitude to the areas of the projections of the parallelogram onto the yz, zx, and xy planes, respectively.

Seven dimensions

Main article: Seven-dimensional cross product

For a and b as vectors in ℝ7, Lagrange's identity takes on the same form as in the case of ℝ

3 [8]

However, the cross product in 7 dimensions does not share all the properties of the cross product in 3 dimensions. For example, the direction of a × b in 7-dimensions may be

the same as c × d even though c and d are linearly independent of a and b. Also the seven dimensional cross product is not compatible with the Jacobi identity.[8]

Quaternions

A quaternion p is defined as the sum of a scalar t and a vector v:

The product of two quaternions p = t + v and q = s + w is defined by

The quaternionic conjugate of q is defined by

and the norm squared is

The multiplicativity of the norm in the quaternion algebra provides, for quaternions p and q:[9]

The quaternions p and q are called imaginary if their scalar part is zero; equivalently, if

Lagrange's identity is just the multiplicativity of the norm of imaginary quaternions,

since, by definition,

Proof of algebraic form

The vector form follows from the Binet-Cauchy identity by setting ci = ai and di = bi. The second version follows by letting ci and di denote the complex conjugates of ai and

bi, respectively,

Here is also a direct proof.[10]

The expansion of the first term on the left side is:

(1)   

which means that the product of a column of as and a row of bs yields (a sum of elements of) a square of abs, which can be broken up into a diagonal and a pair of triangles on

either side of the diagonal.

The second term on the left side of Lagrange's identity can be expanded as:

(2)   

which means that a symmetric square can be broken up into its diagonal and a pair of equal triangles on either side of the diagonal.

To expand the summation on the right side of Lagrange's identity, first expand the square within the summation:

Distribute the summation on the right side,

Now exchange the indices i and j of the second term on the right side, and permute the b factors of the third term, yielding:

(3)   

Back to the left side of Lagrange's identity: it has two terms, given in expanded form by Equations (1) and (2). The first term on the right side of Equation (2) ends up

canceling out the first term on the right side of Equation (1), yielding

(1) - (2) =

which is the same as Equation (3), so Lagrange's identity is indeed an identity, Q.E.D..

See also

Brahmagupta–Fibonacci identity

Lagrange's identity (boundary value problem)

Binet–Cauchy identity

References

^ Eric W. Weisstein (2003). CRC concise encyclopedia of mathematics (http://books.google.com/?id=8LmCzWQYh_UC&pg=PA228) (2nd ed.). CRC Press. ISBN 1584883472.

http://books.google.com/?id=8LmCzWQYh_UC&pg=PA228.

1.

^ Robert E Greene and Steven G Krantz (2006). "Exercise 16" (http://www.amazon.com/Function-Complex-Variable-Graduate-Mathematics/dp/082182905X/ref=sr_1_1?ie=UTF8&

s=books&qid=1271907834&sr=1-1#reader_082182905X) . Function theory of one complex variable (3rd ed.). American Mathematical Society. p. 22. ISBN 0821839624.

http://www.amazon.com/Function-Complex-Variable-Graduate-Mathematics/dp/082182905X/ref=sr_1_1?ie=UTF8&s=books&qid=1271907834&sr=1-1#reader_082182905X.

2.

^ Vladimir A. Boichenko, Gennadiĭ Alekseevich Leonov, Volker Reitmann (2005). Dimension theory for ordinary differential equations (http://books.google.com/?id=9bN1-

b_dSYsC&pg=PA26) . Vieweg+Teubner Verlag. p. 26. ISBN 3519004372. http://books.google.com/?id=9bN1-b_dSYsC&pg=PA26.

3.

^ J. Michael Steele (2004). "Exercise 4.4: Lagrange's identity for complex numbers" (http://books.google.com/?id=bvgBdZKEYAEC&pg=PA68) . The Cauchy-Schwarz master class:

an introduction to the art of mathematical inequalities. Cambridge University Press. pp. 68–69. ISBN 052154677X. http://books.google.com/?id=bvgBdZKEYAEC&pg=PA68.

4.

^ Greene, Robert E.; Krantz, Steven G. (2002). Function Theory of One Complex Variable. Providence, R.I.: American Mathematical Society. p. 22, Exercise 16.

ISBN 978-0-8218-2905-9;

Palka, Bruce P. (1991). An Introduction to Complex Function Theory. Berlin, New York: Springer-Verlag. p. 27, Exercise 4.22. ISBN 978-0-387-97427-9.

5.

^ Howard Anton, Chris Rorres (2010). "Relationships between dot and cross products" (http://books.google.com/?id=1PJ-WHepeBsC&pg=PA162&

dq=%22cross+product%22+%22Lagrange%27s+identity%22&cd=6#v=onepage&q=%22cross%20product%22%20%22Lagrange%27s%20identity%22) . Elementary Linear Algebra:

Applications Version (10th ed.). John Wiley and Sons. p. 162. ISBN 0470432055. http://books.google.com/?id=1PJ-WHepeBsC&pg=PA162&

dq=%22cross+product%22+%22Lagrange%27s+identity%22&cd=6#v=onepage&q=%22cross%20product%22%20%22Lagrange%27s%20identity%22.

6.

^ Pertti Lounesto (2001). Clifford algebras and spinors (http://books.google.com/?id=kOsybQWDK4oC&pg=PA94&

dq=%22which+in+coordinate+form+means+Lagrange%27s+identity%22&cd=1#v=onepage&q=%22which%20in%20coordinate%20form%20means%20Lagrange%27s%20identity%22)

(2nd ed.). Cambridge University Press. p. 94. ISBN 0521005515. http://books.google.com/?id=kOsybQWDK4oC&pg=PA94&

7.

dq=%22which+in+coordinate+form+means+Lagrange%27s+identity%22&cd=1#v=onepage&q=%22which%20in%20coordinate%20form%20means%20Lagrange%27s%20identity%22.

^ a b Door Pertti Lounesto (2001). Clifford algebras and spinors (http://books.google.com/?id=kOsybQWDK4oC&printsec=frontcover&q=Pythagorean) (2nd ed.). Cambridge

University Press. ISBN 0521005515. http://books.google.com/?id=kOsybQWDK4oC&printsec=frontcover&q=Pythagorean. See particularly § 7.4 Cross products in ℝ7

(http://books.google.be/books?id=kOsybQWDK4oC&pg=PA96#v=onepage&q&f=false) , p. 96.

8.

^ Jack B. Kuipers (2002). "§5.6 The norm" (http://books.google.com/?id=_2sS4mC0p-EC&pg=PA111) . Quaternions and rotation sequences: a primer with applications to orbits.

Princeton University Press. p. 111. ISBN 0691102988. http://books.google.com/?id=_2sS4mC0p-EC&pg=PA111.

9.

^ See, for example, Frank Jones, Rice University (http://docs.google.com/viewer?a=v&q=cache:rDnOA-ZKljkJ:www.owlnet.rice.edu/~fjones

/chap7.pdf+lagrange%27s+identity+in+the+seven+dimensional+cross+product&hl=en&gl=ph&sig=AHIEtbQQtdVGhgbYhz78SQQb2biLxRi4kA) , page 4 in Chapter 7 of a book still to

be published (http://www.owlnet.rice.edu/~fjones/) .

10.

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