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I have a book that goes though a detailed development of the following:

[tex]\left[\begin{matrix}J^TJ & J^TV\\ VJ & D \end{matrix}\right]\left[\begin{matrix}p_1 \\ p_2\end{matrix}\right] = \left[\begin{matrix}J^Tr\\ Vr + Ds \end{matrix}\right][/tex]

where V and D are diagonal matrices and everything is known except [itex]p_1[/itex] and [itex]p_2[/itex].

Then it says "since the lower right submatrix D is diagonal, it is easy to eliminate [itex]p_2[/itex] from this system and obtain a smaller n x n system to be solved for [itex]p_1[/itex] alone." The implication is that it's so easy, explanation isn't needed. However, I don't see it.

Can someone explain how to eliminate [itex]p_2[/itex] given D is diagonal?

I've managed to come up with:

[tex][VJ(J^TJ)^{-1}J^TV - D]p_2 = VJ(J^TJ)^{-1}J^Tr - Vr - Ds[/tex]

but this seems far more complicated than what is implied and I don't see how D being diagonal simplifies anything.

[tex]\left[\begin{matrix}J^TJ & J^TV\\ VJ & D \end{matrix}\right]\left[\begin{matrix}p_1 \\ p_2\end{matrix}\right] = \left[\begin{matrix}J^Tr\\ Vr + Ds \end{matrix}\right][/tex]

where V and D are diagonal matrices and everything is known except [itex]p_1[/itex] and [itex]p_2[/itex].

Then it says "since the lower right submatrix D is diagonal, it is easy to eliminate [itex]p_2[/itex] from this system and obtain a smaller n x n system to be solved for [itex]p_1[/itex] alone." The implication is that it's so easy, explanation isn't needed. However, I don't see it.

Can someone explain how to eliminate [itex]p_2[/itex] given D is diagonal?

I've managed to come up with:

[tex][VJ(J^TJ)^{-1}J^TV - D]p_2 = VJ(J^TJ)^{-1}J^Tr - Vr - Ds[/tex]

but this seems far more complicated than what is implied and I don't see how D being diagonal simplifies anything.

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