Final answer:
To construct the normal equations for the least-squares solution of Ax = b, multiply both sides of the equation by the transpose of matrix A: (A^T*A)x = A^T*b.
Step-by-step explanation:
The normal equations for the least-squares solution of Ax = b can be constructed by multiplying both sides of the equation by the transpose of matrix A:
(ATA)x = ATb
where AT is the transpose of matrix A.
The normal equations are derived by setting the derivative of the error function with respect to
�
x equal to zero:
�
�
�
(
∣
∣
�
∣
∣
2
)
=
�
�
�
(
�
�
�
)
=
0
dx
d
(∣∣e∣∣
2
)=
dx
d
(e
T
e)=0
Expanding the expression yields the normal equations:
�
�
�
�
=
�
�
�
A
T
Ax=A
T
b
Where:
�
�
A
T
is the transpose of matrix
�
A.
�
�
�
A
T
A is an
�
×
�
n×n square matrix.
�
�
�
A
T
b is an
�
×
1
n×1 vector.
These normal equations can be solved to find the least-squares solution for the system
�
�
=
�
Ax=b.