Function: qfgaussred
Section: linear_algebra
C-Name: qfgaussred
Prototype: G
Help: qfgaussred(q): square reduction of the (symmetric) matrix q (returns a
square matrix whose i-th diagonal term is the coefficient of the i-th square
in which the coefficient of the i-th variable is 1).
Doc:
\idx{decomposition into squares} of the
quadratic form represented by the symmetric matrix $q$. The result is a
matrix whose diagonal entries are the coefficients of the squares, and the
off-diagonal entries on each line represent the bilinear forms. More
precisely, if $(a_{ij})$ denotes the output, one has
$$ q(x) = \sum_i a_{ii} (x_i + \sum_{j \neq i} a_{ij} x_j)^2 $$
\bprog
? qfgaussred([0,1;1,0])
%1 =
[1/2 1]
[-1 -1/2]
@eprog\noindent This means that $2xy = (1/2)(x+y)^2 - (1/2)(x-y)^2$.
Singular matrices are supported, in which case some diagonal coefficients
will vanish:
\bprog
? qfgaussred([1,1;1,1])
%1 =
[1 1]
[1 0]
@eprog\noindent This means that $x^2 + 2xy + y^2 = (x+y)^2$.
Variant: \fun{GEN}{qfgaussred_positive}{GEN q} assumes that $q$ is
positive definite and is a little faster; returns \kbd{NULL} if a vector
with negative norm occurs (non positive matrix or too many rounding errors).