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D.14.3.1 signatureL
Procedure from library phindex.lib (see phindex_lib).
- Usage:
- signatureL(M[,r]); M symmetric matrix, r int (optional).
- Return:
- the signature of M of type int or if r is given and !=0 then
intvec with (signature, nr. of +, nr. of -) is returned.
- Theory:
- Given the matrix M, we construct the quadratic form associated. Afterwards
we use the method of Lagrange to compute the signature. The law of
inertia for a real quadratic form A(x,x) says that in a
representation of A(x,x) as a sum of independent squares
A(x,x)=sum_{i=1}^r a_iX_i^2.
The number of positive and the number of negative squares are
independent of the choice of representation. The signature -s- of
A(x,x) is the difference between the number -pi- of positive squares
and the number -nu- of negative squares in the representation of
A(x,x). The rank -r- of M (or A(x,x)) and the signature -s-
determine the numbers -pi- and -nu- uniquely, since
r=pi+nu, s=pi-nu.
The method of Lagrange is a procedure to reduce any real quadratic
form to a sum of squares.
Ref. Gantmacher, The theory of matrices, Vol. I, Chelsea Publishing
Company, NY 1960, page 299.
Example:
| | LIB "phindex.lib";
ring r=0,(x),ds;
matrix M[5][5]=0,0,0,1,0,0,1,0,0,-1,0,0,1,0,0,1,0,0,3,0,0,-1,0,0,1;
signatureL(M,1); //The rank of M is 3+1=4
==> 2,3,1
matrix H[5][5]=0,-7,0,1,0,-7,1,0,0,-1,0,0,1,0,0,1,0,0,-3,5,0,-1,0,5,1;
signatureL(H);
==> 1
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