# Jacobi method

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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 be a given bilinear form (not necessarily symmetric) over a field . Suppose that its matrix satisfies the condition (1)

where is the minor of order in the upper left-hand corner. Then can be written in the form (2)

where , , and for , (3a) (3b)

In particular, if is a symmetric matrix satisfying (1) and is the quadratic form with matrix , then can be reduced to the canonical form (4)

by using the following transformation of the unknowns: (5)

for , and This transformation has a triangular matrix, and can be written as (6)

where is the minor of that stands in the rows , and in the columns .

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

When the matrix of satisfies only the conditions  where is the rank of the form, can be reduced to the canonical form (7)

(here ) by a triangular transformation of the unknowns. This reduction can be realized by using the Gauss method (see ). If, in particular, , then the positive index of inertia of 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 