# Jacobi method

Jump to: navigation, search

A method for reducing a quadratic form (cf. also Quadratic forms, reduction of) to canonical form by using a triangular transformation of the unknowns; it was suggested by C.G.J. Jacobi (1834) (see ).

Let

$$f = \ \sum _ {i, k = 1 } ^ { n } a _ {ki} x _ {i} y _ {k}$$

be a given bilinear form (not necessarily symmetric) over a field $P$. Suppose that its matrix $A = \| a _ {ki} \|$ satisfies the condition

$$\tag{1 } \Delta _ {k} \neq 0,\ \ k = 1 \dots n,$$

where $\Delta _ {k}$ is the minor of order $k$ in the upper left-hand corner. Then $f$ can be written in the form

$$\tag{2 } f = \ \sum _ {k = 1 } ^ { n } \frac{u _ {k} v _ {k} }{\Delta _ {k - 1 } \Delta _ {k} } ,$$

where $u _ {1} = \partial f/ \partial y _ {1}$, $v _ {1} = \partial f/ \partial x _ {1}$, and for $k = 2 \dots n$,

$$\tag{3a } u _ {k} = \ \left \| \begin{array}{cccc} a _ {11} &\dots &a _ {1k - 1 } & \frac{\partial f }{\partial y _ {1} } \\ \dots &\dots &\dots &\dots \\ a _ {k1} &\dots &a _ {kk - 1 } & \frac{\partial f }{\partial y _ {k} } \\ \end{array} \ \right \| ,$$

$$\tag{3b } v _ {k} = \ \left \| \begin{array}{cccc} a _ {11} &\dots &a _ {1k - 1 } & \frac{\partial f }{\partial x _ {1} } \\ \dots &\dots &\dots &\dots \\ a _ {1k} &\dots &a _ {k - 1k } & \frac{\partial f }{\partial x _ {k} } \\ \end{array} \ \right \| .$$

In particular, if $A$ is a symmetric matrix satisfying (1) and $f$ is the quadratic form with matrix $A$, then $f$ can be reduced to the canonical form

$$\tag{4 } f = \ \sum _ {k = 1 } ^ { n } \frac{u _ {k} ^ {2} }{\Delta _ {k - 1 } \Delta _ {k} } ,\ \ \Delta _ {0} = 1,$$

by using the following transformation of the unknowns:

$$\tag{5 } u _ {k} = \ \left \| \begin{array}{cccc} a _ {11} &\dots &a _ {1k - 1 } &{ \frac{1}{2} } \frac{\partial f }{\partial x _ {1} } \\ \dots &\dots &\dots &\dots \\ a _ {k1} &\dots &a _ {kk - 1 } &{ \frac{1}{2} } \frac{\partial f }{\partial x _ {k} } \\ \end{array} \ \right \|$$

for $k = 2 \dots n$, and

$$u _ {1} = \ { \frac{1}{2} } \frac{\partial f }{\partial x _ {1} } .$$

This transformation has a triangular matrix, and can be written as

$$\tag{6 } u _ {k} = \ \sum _ {i = k } ^ { n } C _ {ki} x _ {i} ,$$

where $C _ {ki}$ is the minor of $A$ that stands in the rows $1 \dots k$, and in the columns $1 \dots k - 1, i$.

The formulas (2)–(7) are called Jacobi's formulas.

When the matrix of $f$ satisfies only the conditions

$$\Delta _ {i} \neq 0,\ \ i = 1 \dots r,$$

$$\Delta _ {r + 1 } = \dots = \Delta _ {n} = 0,$$

where $r$ is the rank of the form, $f$ can be reduced to the canonical form

$$\tag{7 } f = \ \sum _ {k = 1 } ^ { r } \frac{u _ {k} ^ {2} }{\Delta _ {k - 1 } \Delta _ {k} }$$

(here $\Delta _ {0} = 1$) by a triangular transformation of the unknowns. This reduction can be realized by using the Gauss method (see ). If, in particular, $P = \mathbf R$, then the positive index of inertia of $f$ is equal to the number of preservations of sign, and the negative index of inertia is equal to the number of changes of sign in the series of numbers

$$1, \Delta _ {1} \dots \Delta _ {r} .$$

See also Law of inertia.

## Contents

How to Cite This Entry:
Jacobi method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_method&oldid=47458
This article was adapted from an original article by I.V. Proskuryakov, G.D. Kim (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article