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Suppose $\mathbf{x}\in R^{m\times 1}$, $\mathbf{X} = [\mathbf{x}\, \mathbf{x}\, \cdots]^\top\in R^{nm\times 1}$ and $\mathbf{b}\in R^{nm\times 1}$ and $\mathbf{A}\in R^{nm\times nm}$. How to solve

\begin{equation} \mathrm{argmin}_{\mathbf{x}} (\mathbf{b}-\mathbf{A}\mathbf{X})^\top(\mathbf{b}-\mathbf{A}\mathbf{X}) \end{equation} ?

Notice that argmin is w.r.t. the small $\mathbf{x}$.

(actually, what is the derivative of that objective function w.r.t. $\mathbf{x}$?)

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  • $\begingroup$ Kronecker products and vectorizations $\endgroup$ – mathreadler Nov 18 at 20:39
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Hint. Consider the bilinear form $\psi(x,y)=(x|y)$ (inner product). Then, $\psi'(x,y)\cdot (h,k)=\psi(x,k)+\psi(h,y).$ Also, the function $x\mapsto X=(x,\ldots,x)$ is linear, thus its derivative at any point is itself. Similarly, define $u(y)=(y,y),$ which is also linear and $v(t)=b-At,$ which has derivative $v'(t) \cdot s=As.$ You want to differentiate $\psi \circ u \circ v \circ X,$ which follows readily from chain rule.

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