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Lagrange's Identity

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Lagrange's identity is the algebraic identity

(sum_(k=1)^na_kb_k)^2=(sum_(k=1)^na_k^2)(sum_(k=1)^nb_k^2)-sum_(1<=k<j<=n)(a_kb_j-a_jb_k)^2
(1)

(Mitrinović 1970, p. 41; Marsden and Tromba 1981, p. 57; Gradshteyn and Ryzhik 2000, p. 1049).

Lagrange's identity is a special case of the Binet-Cauchy identity, and Cauchy's inequality in n dimensions follows from it.

It can be coded in the Wolfram Language as follow. 

  LagrangesIdentity[n_] := Module[
    {aa = Array[a, n], bb = Array[b, n]},
   Total[(aa^2) Plus @@ (bb^2)] ==
   Total[(a[#1]b[#2] - a[#2]b[#1])^2& @@@
      Subsets[Range[n], {2}]] + (aa.bb)^2
  ]

Plugging in gives the n=2 and n=3 identities

(a_1^2+a_2^2)(b_1^2+b_2^2)=(a_1b_1+a_2b_2)^2+(a_1b_2-a_2b_1)^2
(2)
(a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)=(a_1b_1+a_2b_2+a_3b_3)^2
(3)
+[(a_1b_2-a_2b_1)^2+(a_1b_3-a_3b_1)^2+(a_2b_3-a_3b_2)^2].
(4)

A vector quadruple product formula known as Lagrange's identity given by

(axb)·(cxd)=(a·c)(b·d)-(b·c)(a·d)
(5)

(Bronshtein and Semendyayev 2004, p. 185).

A related identity also known as Lagrange's identity is given by defining a_k and b_k to be n-dimensional vectors for k=1, ..., n-1. Then

(a_1×...×a_(n-1))·(b_1×...×b_(n-1))=|a_1·b_1 ... a_1·b_(n-1); | ... |; a_(n-1)·b_1 ... a_(n-1)·b_(n-1)|
(6)

(Greub 1978, p. 155), where AxB denotes a cross product, A·B denotes a dot product, and detA is the determinant of the matrix A.

See also

Binet-Cauchy Identity, Cauchy's Inequality, Vector Triple Product, Vector Quadruple Product

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 32, 1985.Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook of Mathematics, 4th ed. New York: Springer-Verlag, 2004.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.Griffiths, D. J. Introduction to Electrodynamics. Englewood Cliffs, NJ: Prentice-Hall, p. 13, 1981.Greub, W. Multilinear Algebra, 2nd ed. New York: Springer-Verlag, 1978.Marsden, J. E. and Tromba, A. J. Vector Calculus, 2nd ed. New York: W. H. Freeman, 1981.Mitrinović, D. S. Analytic Inequalities. New York: Springer-Verlag, 1970.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 114, 1953.

Referenced on Wolfram|Alpha

Lagrange's Identity

Cite this as:

Weisstein, Eric W. "Lagrange's Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LagrangesIdentity.html

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