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Appendix A – Overdetermined linear equations systems
b −AX ≤ b −AY , ∀Y ∈R
n
(A-15)
Rewriting b − AY as
( ) ( )
b − AY = b − AX + A X −Y
(A-16)
and taking the norm, we obtain
2
2
( ) ( )
T
(( b AX) A( X Y)
) (( b AX) A( X Y)
)
T
( b AX) 2( A( X Y)
) ( b AX) A( X Y)
b − AY = b − AX + A X −Y
= − + − − + −
2 2
= − + − − + −
(A-17)
tel-00623090, version 1 - 13 Sep 2011
Moreover, by (A-9) we have
Therefore, we conclude:
T T
T
( A( X −Y)
) ( b − AX) = ( X −Y) A ( b − AX) = 0
2 2 2 2
( ) ( ) ( )
b − AY = b − AX + A X −Y ≥ b −AX
The equality in (A-19) only occurs when A( X − Y) = 0
of the matrix A are linearly independent, then A( X − Y) = 0
that ( X − Y) = 0
, i.e., X = Y .
(A-18)
(A-19)
. Furthermore, once the columns
which implies
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