Section 22

Least squares

avanzadoviz

\hat x=\arg\min_x\|Ax-b\|_2^2

Seeks the best approximation when Ax=b has no exact solution.

avanzadoviz

A^TA\hat x=A^Tb

Characterize the projection of b onto the column space of A.

The closed-form expression $(A^TA)^{-1}A^Tb$ requires the columns of $A$ to be linearly independent. For general cases, QR or SVD is preferred.

avanzadoviz

\hat x=A^+b;\quad A^+=V\Sigma^+U^T

Produces minimum-norm least-squares solutions.

Loading section…

Section 22

avanzadoviz

\hat x=\arg\min_x\|Ax-b\|_2^2

Seeks the best approximation when Ax=b has no exact solution.

avanzadoviz

A^TA\hat x=A^Tb

Characterize the projection of b onto the column space of A.

The closed-form expression $(A^TA)^{-1}A^Tb$ requires the columns of $A$ to be linearly independent. For general cases, QR or SVD is preferred.

avanzadoviz

\hat x=A^+b;\quad A^+=V\Sigma^+U^T

Produces minimum-norm least-squares solutions.