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Function QXA
Function QXA takes as arguments a quadratic form in stack level 2 and a
vector of variables in stack level 1, returning the square matrix A from which
the quadratic form is derived in stack level 2, and the list of variables in stack
level 1. For example,
'X^2+Y^2-Z^2+4*X*Y-16*X*Z' `
['X','Y','Z'] ` QXA
returns
2: [[1 2 –8][2 1 0][-8 0 –1]]
1: [‘X’ ‘Y’ ‘Z’]
Diagonal representation of a quadratic form
Given a symmetric square matrix A, it is possible to “diagonalize” the matrix
A by finding an orthogonal matrix P such that P
T
⋅A⋅P = D, where D is a
diagonal matrix. If Q = x⋅A⋅x
T
is a quadratic form based on A, it is possible
to write the quadratic form Q so that it only contains square terms from a
variable y, such that x = P⋅y, by using Q = x⋅A⋅x
T
= (P⋅y)⋅A⋅ (P⋅y)
T
=
y⋅(P
T
⋅A⋅P)⋅y
T
= y⋅D⋅y
T
.
Function SYLVESTER
Function SYLVESTER takes as argument a symmetric square matrix A and
returns a vector containing the diagonal terms of a diagonal matrix D, and a
matrix P, so that P
T
⋅A⋅P = D. For example,
[[2,1,-1],[1,4,2],[-1,2,-1]] SYLVESTER
produces
2: [ 1/2 2/7 -23/7]
1: [[2 1 –1][0 7/2 5/2][0 0 1]]
Function GAUSS
Function GAUSS returns the diagonal representation of a quadratic form Q =
x⋅A⋅x
T
taking as arguments the quadratic form in stack level 2 and the vector
of variables in stack level 1. The result of this function call is the following:
• An array of coefficients representing the diagonal terms of D (stack
level 4)
• A matrix P such that A = P
T
⋅D⋅P (stack level 3)
• The diagonalized quadratic form (stack level 2)
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