Background
I have been studying general linear model which involves many operations of block matrices. I encounter the following equality, which I cannot derive spending two hours. I think I must have overlooked some facts about block matrices:
Question
Let $\epsilon$ and $\beta$ be a vector in $\mathbb{R}^n$, and $X$ be a matrix of dimension $n\times d $, and it is of full column rank. Let $A$ be a matrix of dimension $q \times d$ of full row rank and $B$ be a matrix of dimension $(r-q) \times d$ of full row rank. Let $a$ be a vector in $\mathbb{R}^q$ and $b$ be a vector in $\mathbb{R}^{r-q}$, prove that
$-\left[\begin{array}{c}\boldsymbol{a}-\boldsymbol{A \beta} \\ \boldsymbol{b}-\boldsymbol{B} \boldsymbol{\beta}\end{array}\right]^{\top}\left\{\left[\begin{array}{c}\boldsymbol{A} \\ \boldsymbol{B}\end{array}\right]\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1}\left[\boldsymbol{A}^{\top}, \boldsymbol{B}^{\top}\right]\right\}^{-1}$ $\cdot\left\{2\left[\begin{array}{c}\boldsymbol{A} \\ \boldsymbol{B}\end{array}\right]\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{\epsilon}-\left[\begin{array}{c}\boldsymbol{a}-\boldsymbol{A \beta} \\ \boldsymbol{b}-\boldsymbol{B} \boldsymbol{\beta}\end{array}\right]\right\} + (\boldsymbol{a}-\boldsymbol{A} \boldsymbol{\beta})^{\top}\left\{\boldsymbol{A}\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{A}^{\top}\right\}^{-1}\left\{2 \boldsymbol{A}\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{\epsilon}-\boldsymbol{a}+\boldsymbol{A} \boldsymbol{\beta}\right\}$
$=-\left[\begin{array}{c}\mathbf{0}_{q} \\ \boldsymbol{b}-\boldsymbol{B} \boldsymbol{\beta}\end{array}\right]^{\top}\left\{\left[\begin{array}{l}\boldsymbol{A} \\ \boldsymbol{B}\end{array}\right]\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1}\left[\boldsymbol{A}^{\top}, \boldsymbol{B}^{\top}\right]\right\}^{-1}$ $\left\{2\left[\begin{array}{c}\boldsymbol{A} \\ \boldsymbol{B}\end{array}\right]\left(\boldsymbol{X}^{\top} \boldsymbol{X}\right)^{-1} \boldsymbol{X}^{\top} \boldsymbol{\epsilon}-\left[\begin{array}{c}\boldsymbol{0}_{q} \\ \boldsymbol{b}-\boldsymbol{B} \boldsymbol{\beta}\end{array}\right]\right\}$
Remarks
As you can see, this expression is complicated. The major difficulty I have is with the dimension of the terms on the right side of the addition in LHS. This term's components do not have the same dimension as the terms to the left of the addition. But because this term, when considered on its own is a scalar, the addition is still legitimate. But how can such terms be reduced to the one on RHS?