Etude de la combustion de gaz de synthèse issus d'un processus de ...

Appendix A – Overdetermined linear equations systems
This is a system of n equations and n unknowns for which a solution is expected.
In order to prove the existence and unicity of solution of the normal equations (A-10) it
is necessary to prove that: The matrix
matrix A are linearly independent.
T
A Ais invertible if and only if the columns of the
If the columns of A would be linearly independent, then it implies that AX ≠ 0 .
Therefore, we have:
2
A X = AX AX = X A A X > 0,
∀X
≠ 0
T
T T
( ) ( ) ( )
(A-11)
which means that
T
AA is a symmetric positive definite matrix, consequently invertible.
tel-00623090, version 1 - 13 Sep 2011
Let us now assume that
T
such that A AX = 0 , i.e.,
T
A A it is not invertible. In this case there exits a vector X ≠ 0
2
T T
T
( ) ( )
X A AX = 0⇒ AX AX = AX = 0⇒ AX = 0
(A-12)
Therefore, we concluded that the columns of A are not linearly independent.
The existence and unicity of a solution of the normal equations depend on the linear
independence of the columns matrix associated. In the case of curve fitting, where the
curve P takes the form
n
j = 1
( )
Px ( ) = ∑ au
j j
x
(A-13)
we have that the elements of the matrix A are given by
( )
A = u x
(A-14)
ij j i
Therefore, the linear independence of the columns of A depends on the functions u j
and on the number and localization of the points x i , In general, it is not easy to know a
priori if that linear independence is verified.
At this point we guaranty the existence and unicity of solutions of normal equations (A-
10). Our next goal is to prove that this solution gives us the minimum residue, i.e., the
solution X of the equation (A-10) satisfies the relation
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