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So I have a problem in the form of $y = Ax$, where $A$ is a matrix and $x$ and $y$ are vectors. $A$ is a skinny matrix and I would like to do a least squares solution to solve for $x$. Let's say though that $x$ has $10$ rows. In my solution I require that the first say $3$ elements of $x = [A;B;C]$. Does anyone have any thoughts on how I can perform a least squares solution with the constraint that the first $3$ rows of my solution are already defined?

Thanks!

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Partition $A$ as $A=\begin{bmatrix} A_1 & A_2\end{bmatrix}$, and $x = (x_1, x_2)^T$. Then $Ax = A_1 x_1 +A_2 x_2$, and the problem becomes $\min_{x_2} \|Ax-y\|^2 = \min_{x_2} \|A_2x_2-(y-A_1 x_1) \|^2$, that is, an unconstrained least squares problem over $x_2$ with the $y$ vector replaced by $y-A_1 x_1$.

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  • $\begingroup$ Thanks, so much for your help. This makes sense! $\endgroup$
    – PumpkinPie
    Oct 23 '14 at 5:35

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