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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)
• The list of variables (stack level 1)
For example:
'X^2+Y^2-Z^2+4*X*Y-16*X*Z' `
['X','Y','Z'] ` GUSS
returns
4: [1 –0.333 20.333]
3: [[1 2 –8][0 –3 16][0 0 1]]
2: ’61/3*Z^2+ -1/3*(16*Z+-3*Y)^2+(-8*z+2*Y+X)^2‘
1: [‘X’ ‘Y’ ‘Z’]
Linear Applications
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