Lagrange method

A method for reducing a quadratic form (cf. also Quadratic forms, reduction of) to a sum of squares, given by J.L. Lagrange in 1759. Suppose one is given a quadratic form

$$ \tag{1 } f ( x) = \sum _ {i, j = 1 } ^ { n } a _ {ij} x _ {i} x _ {j} ,\ a _ {ij} = a _ {ji} , $$

in $ n $ variables $ x _ {1} \dots x _ {n} $, with coefficients from a field $ k $ of characteristic $ \neq 2 $. It is required to reduce this form to the canonical form

$$ \tag{2 } f ( x) = \sum _ { i= } 1 ^ { r } b _ {i} y _ {i} ^ {2} ,\ b _ {i} \in k ,\ b _ {i} \neq 0 ,\ i= 1 \dots r , $$

by means of a non-singular linear transformation of the variables. Lagrange's method consists in the following. One may assume that not all the coefficients in (1) are zero. Therefore, two cases are possible.

1) For some $ g $, $ 1 \leq g \leq n $, the diagonal coefficient $ a _ {gg} \neq 0 $. Then

$$ \tag{3 } f ( x) = \frac{1}{a _ {gg} } \left ( \sum _ { k= } 1 ^ { n } a _ {gk} x _ {k} \right ) ^ {2} + f _ {1} ( x) , $$

where the form $ f _ {1} ( x) $ does not contain the variable $ x _ {g} $.

2) If all the $ a _ {gg} = a _ {hh} = 0 $, but $ a _ {gh} \neq 0 $, then

$$ \tag{4 } f ( x) = \frac{1}{2a _ {gh} } \left [ \sum _ { k= } 1 ^ { n } ( a _ {gk} + a _ {hk} ) x _ {k} \right ] ^ {2} + $$

$$ - \frac{1}{2a _ {gk} } \left [ \sum _ { k= } 1 ^ { n } ( a _ {gk} - a _ {hk} ) x _ {k} \right ] ^ {2} + f _ {2} ( x) , $$

where the form $ f _ {2} ( x) $ does not contain the two variables $ x _ {g} $ and $ x _ {h} $. The forms in square brackets in (4) are linearly independent. By using transformations of the form (3) and (4), after finitely many steps one can reduce the form (1) to a sum of squares of linearly independent linear forms. By means of partial derivatives the formulas (3) and (4) can be written as

$$ f ( x) = \frac{1}{4a _ {gg} } \left ( \frac{\partial f }{\partial x _ {g} } \right ) ^ {2} + f _ {1} ( x) , $$

$$ f ( x) = \frac{1}{8a _ {gh} } \left [ \left ( \frac{\partial f }{\partial x _ {g} } + \frac{\partial f }{\partial x _ {h} } \right ) ^ {2} - \left ( \frac{ \partial f }{\partial x _ {g} } - \frac{\partial f }{\partial x _ {h} } \right ) ^ {2} \right ] + f _ {2} ( x) . $$

References

[1]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian) MR1657129 MR0107649 MR0107648 Zbl 0927.15002 Zbl 0927.15001 Zbl 0085.01001
[2]	A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian) MR0945393 MR0926059 MR0778202 MR0759341 MR0628003 MR0384363 Zbl 0237.13001
[3]	P.S. Aleksandrov, "Lectures on analytical geometry" , Moscow (1968) (In Russian) MR0244836

Comments

See also Law of inertia.