Lagrange\'s identity

In mathematics, Lagrange's identity is the algebraic equation

$\biggl( \sum_{k=1}^n a_k^2\biggr) \biggl(\sum_{k=1}^n b_k^2\biggr) - \biggl(\sum_{k=1}^n a_k b_k\biggr)^2 = \sum_{i=1}^{n-1} \sum_{j=i+1}^n (a_i b_j - a_j b_i)^2 \biggl(= {1 \over 2} \sum_{i=1}^n \sum_{j=1}^n (a_i b_j - a_j b_i)^2\biggr),$

which applies to any two sets {a₁, a₂, . Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, an algebraic equation over a given field is an Equation of the form P = Q where P and Q . . , a_n} and {b₁, b₂, . . . , b_n} of real or complex numbers (or more generally, elements of a commutative ring). In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property This identity is a special form of the Binet–Cauchy identity. In Algebra, the Binet–Cauchy identity named after Jacques Philippe Marie Binet and Augustin Louis Cauchy, states that \biggl(\sum_{i=1}^n For complex numbers it can also be written in the form

$\biggl( \sum_{k=1}^n |a_k|^2\biggr) \biggl(\sum_{k=1}^n |b_k|^2\biggr) - \biggl|\sum_{k=1}^n a_k b_k\biggr|^2 = \sum_{i=1}^{n-1} \sum_{j=i+1}^n |a_i b_j - a_j b_i|^2$

involving the absolute value. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

Since the right-hand side of the identity is clearly non-negative, it implies Cauchy's inequality in the finite-dimensional real coordinate space $\mathbb{R}^n$ and its complex counterpart $\mathbb{C}^n$ . In Mathematics, the Cauchy–Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy–Schwarz–Bunyakovsky In Mathematics, the dimension of a Vector space V is the cardinality (i

1 Lagrange's identity and exterior algebra
2 Lagrange's identity and vector calculus
3 Lagrange's identity and calculus
4 Proof
- 4.1 Algebraic form
- 4.2 Calculus form^[1]
5 See also
6 References

Lagrange's identity and exterior algebra

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

$(a \cdot a)(b \cdot b) - (a \cdot b)^2 = (a \wedge b) \cdot (a \wedge b).$

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

$\|a \wedge b\| = \sqrt{(\|a\|\ \|b\|)^2 - \|a \cdot b\|^2}.$

Lagrange's identity and vector calculus

If a and b are vectors in $\mathbb{R}^3$ , Lagrange's identity can be also written in terms of the cross product and dot product:

$|a \times b|^2 + (a \cdot b)^2 = |a|^2 |b|^2 = (a \cdot a)(b \cdot b).$

This is a special case of the multiplicativity of the norm in the quaternion algebra:

$|vw| = |v| |w|. \,$

Or more generally,

$(v \times w) \cdot (a \times b) = (v \cdot a)(w \cdot b) - (v \cdot b)(w \cdot a)$

Lagrange's identity and calculus

In terms of the Sturm-Liouville theory, Lagrange's identity can be written

$\int_0^1(Lu)v-u(Lv)\,dx=-p(u'v-uv')\bigg\|_0^1$ ^[1]		$(1)$

where $p = P (x)$ , $q = Q (x)$ , $u = U (x)$ and $v = V (x)$ are functions of $x$ . In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics and its applications a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855 and Joseph Liouville $u$ and $v$ having continuous second derivatives on the interval $[0,1]$ . $L$ is Sturm-Liouville differential operators defined by

$L u = - (p u')' + q u$		$(2)$

Proof

Algebraic form

The first version follows from the Binet-Cauchy identity by setting c_i = a_i and d_i = b_i. In Mathematics and its applications a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855 and Joseph Liouville The second version follows by letting c_i and d_i denote the complex conjugates of a_i and b_i, respectively,

Here is also a direct proof of the first version. In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. The expansion of the first term on the left side is

$\left( \sum_{k=1}^n a_k^2\right) \left(\sum_{k=1}^n b_k^2\right) = \sum_{i=1}^n \sum_{j=1}^n a_i^2 b_j^2 = \sum_{k=1}^n a_k^2 b_k^2 + \sum_{i=1}^{n-1} \sum_{j=i+1}^n a_i^2 b_j^2 + \sum_{j=1}^{n-1} \sum_{i=j+1}^n a_i^2 b_j^2$		$(3)$

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 like so

$\left(\sum_{k=1}^n a_k b_k\right)^2 = \sum_{k=1}^n a_k^2 b_k^2 + 2\sum_{i=1}^{n-1} \sum_{j=i+1}^n a_i b_i a_j b_j$		$(4)$

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:

$\sum_{i=1}^{n-1} \sum_{j=i+1}^n (a_i b_j - a_j b_i)^2 = \sum_{i=1}^{n-1} \sum_{j=i+1}^n (a_i^2 b_j^2 + a_j^2 b_i^2 - 2 a_i b_j a_j b_i).$

Distribute the summation on the right side,

$\sum_{i=1}^{n-1} \sum_{j=i+1}^n (a_i b_j - a_j b_i)^2 = \sum_{i=1}^{n-1} \sum_{j=i+1}^n a_i^2 b_j^2 + \sum_{i=1}^{n-1} \sum_{j=i+1}^n a_j^2 b_i^2 - 2 \sum_{i=1}^{n-1} \sum_{j=i+1}^n a_i b_j a_j b_i .$

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

$\sum_{i=1}^{n-1} \sum_{j=i+1}^n (a_i b_j - a_j b_i)^2 = \sum_{i=1}^{n-1} \sum_{j=i+1}^n a_i^2 b_j^2 + \sum_{j=1}^{n-1} \sum_{i=j+1}^n a_i^2 b_j^2 - 2 \sum_{i=1}^{n-1} \sum_{j=i+1}^n a_i b_i a_j b_j .$		$(5)$

Back to the left side of Lagrange's identity: it has two terms, given in expanded form by Equations (3) and (4). The first term on the right side of Equation (4) ends up cancelling out the first term on the right side of Equation (3), yielding

$(3) - (4) = \sum_{i=1}^{n-1} \sum_{j=i+1}^n a_i^2 b_j^2 + \sum_{j=1}^{n-1} \sum_{i=j+1}^n a_i^2 b_j^2 - 2\sum_{i=1}^{n-1} \sum_{j=i+1}^n a_i b_i a_j b_j$

which is the same as Equation (5), so Lagrange's identity is indeed an identity, q. e. d.. QED is an abbreviation of the Latin phrase "la '''quod erat demonstrandum'''" which means literally "that which was to be demonstrated"

Calculus form^[1]

Replace $f (x) = p u'$ , $g (x) = v$ , $a = 0$ and $b = 1$ into the rule integration by parts

$\int_a^bf'(x)\,g(x)\,dx=f(x)\,g(x)\bigg\|_a^b-\int_a^bf(x)\,g'(x)\,dx$		$(6)$

we have

$\int_0^1(pu')'v\,dx=(pu')v\bigg\|_0^1-\int_0^1(pu')v'\,dx=p(u'v)\bigg\|_0^1-\int_0^1(pu')v'\,dx$		$(7)$

Replace $f (x) = u$ , $g (x) = p v'$ , $a = 0$ and $b = 1$ into the rule (6) again, we have

$\int_0^1u'(pv')\,dx=u(pv')\bigg|_0^1-\int_0^1u(pv')'\,dx$

$-\int_0^1(pu')v'\,dx=-p(uv')\bigg\|_0^1+\int_0^1u(pv')'\,dx$		$(8)$

Replace (8) into (7), we get

$\int_0^1(pu')'v\,dx=p(u'v)\bigg|_0^1-p(uv')\bigg|_0^1+\int_0^1u(pv')'\,dx$

$-\int_0^1(pu')'v\,dx=-p(u'v-uv')\bigg\|_0^1-\int_0^1u(pv')'\,dx$		$(9)$

From the definition (2), we can get

$\int_0^1(Lu)v\,dx=\int_0^1[-(pu')'+qu]v\,dx=-\int_0^1(pu')'v\,dx+\int_0^1uqv\,dx$		$(10)$

Replace (9) into (10), we have

$\int_0^1(Lu)v\,dx=-p(u'v-uv')\bigg|_0^1-\int_0^1u(pv')'\,dx+\int_0^1uqv\,dx$

$\int_0^1(Lu)v\,dx=-p(u'v-uv')\bigg\|_0^1+\int_0^1u(Lv)\,dx$		$(11)$

Rearrange terms of (11) then (1) is obtained. q. e. d.

References

^ ^a ^b Boyce, William E. ; Richard C. DiPrima (2001). "Boundary Value Problems and Sturm–Liouville Theory", Elementary Differential Equations and Boundary Value Problems (PDF), 7th ed. (in English), New York: John Wiley & Sons, pp. 630. ISBN 0-471-31999-6. OCLC 64431691. The OCLC Online Computer Library Center is according to its website a "nonprofit membership computer library service and research organization dedicated to the public purpose (for the two sections Lagrange's identity and calculus and Calculus form of this article)

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