# Example 1.1
### a)

\begin{align}
\frac{1}{2}(x-v)^TH(x-v)+d &= \frac{1}{2}(x-a)^TA(x-a) + \frac{1}{2}(x-b)^TB(x-b) \\[12pt]
x^THx - 2x^THv + v^THv + d &= x^TAx - 2x^TAa + a^TAa + x^TBx - 2x^TBb + b^TBb \\[12pt]
&= x^T(A+B)x - 2x^T(Aa+Bb) + a^TAa + b^TBb
\end{align}

#
\begin{align}
H &= A+B \\[6pt]
v &= H^{-1}(Aa+Bb) \\[6pt]
d &= -v^THv + a^TAa + b^TBb\\[6pt]
&=-(Aa+Bb)^TH^{-1}(Aa+Bb) + a^TAa + b^TBb
\end{align}

#
### b)
\begin{align}
\frac{1}{2}(x-v)^TH(x-v)+d &= \frac{1}{2}(x-a)^TA(x-a) + \frac{1}{2}(Cx-b)^TB(Cx-b) \\[12pt]
x^THx - 2x^THv + v^THv + d &= x^TAx - 2x^TAa + a^TAa + x^TC^TBCx - 2x^TC^TBb + b^TBb \\[12pt]
&= x^T(A+C^TBC)x - 2x^T(Aa+C^TBb) + a^TAa + b^TBb
\end{align}

#
\begin{align}
H &= A+C^TBC \\[6pt]
v &= H^{-1}(Aa+C^TBb) \\[6pt]
d &= -v^THv + a^TAa + b^TBb\\[6pt]
&=-(Aa+C^TBb)^TH^{-1}(Aa+C^TBb) + a^TAa + b^TBb
\end{align}

#
### c)
lemma 1.54: $(A+BCD)^{-1} = A^{-1} - A^{-1}B(DA^{-1}B+C^{-1})^{-1}DA^{-1}$
lemma 1.55: $(A+BCD)^{-1}BC = A^{-1}B(DA^{-1}B+C^{-1})^{-1}$

#
\begin{align}
d &= \overline{b}^T(B - B^TCH^{-1}C^TB)\overline{b} \\[6pt]
&= \overline{b}^T(B - B^TC(A + C^TBC)^{-1}C^TB)\overline{b} \\[6pt]
&= \overline{b}^T(CA^{-1}C^T + B^{-1})^{-1}\overline{b}
\end{align}

#
\begin{align}
&\frac{1}{2}\Big[(\overline{x}-\overline{v})^TH(\overline{x}-\overline{v})+d\Big] = \frac{1}{2}\Big[(x-a-\overline{v})^T\widetilde{H}^{-1}(x-a-\overline{v})+d\Big] \\[6pt]
&= \frac{1}{2}\Big[(x-a-A^{-1}C^T(CA^{-1}C^T + B^{-1})^{-1}\overline{b})^T\widetilde{H}^{-1}(x-a-A^{-1}C^T(CA^{-1}C^T + B^{-1})^{-1}\overline{b})+d\Big] \\[6pt]
&= \frac{1}{2}\Big[(x-v)^T\widetilde{H}^{-1}(x-v)+d\Big]
\end{align}