How these denominators arose?

Question

$$ p_{ij}=p_{ji} ~\text{is held for positive integers} ~~~i,j \tag{1} $$

$$ V_{1} = p_{11}Q_{1} + p_{12} Q_{2} + p_{13} Q_{3} + \cdot\cdot\cdot \tag{2}$$

$$ V_{2} = p_{21}Q_{1} + p_{22} Q_{2} + p_{23} Q_{3} + \cdot\cdot\cdot \tag{3}$$

$$ V_{1} =V_{2} = V \tag{4}$$

$$ Q_{1} +Q_{2} =Q \tag{5}$$

What I can't get currently are the equations of $~Q_{1} ~,~ V_{3}$

$$ Q_{1} =\frac{p_{22} - p_{12} }{p_{11}+ p_{22} -2 p_{12} } Q+ \frac{\left( p_{23} -p_{13} \right) Q_{3} +\left( p_{24} -p_{14} \right) Q_{4} + \cdot\cdot\cdot }{p_{11}+ p_{22} -2 p_{12} } \tag{6}$$

$$ \therefore ~~ V_{3} =p_{31} Q_{1} + p_{32} Q_{2} +p_{33} Q_{3} + \cdot\cdot\cdot $$

$$ = \left( p_{31} - p_{32} \right) Q_{1} +p_{32} Q + p_{33} Q_{3} + \cdot\cdot\cdot $$

$$ = \left\{ p_{32} + \frac{\left( p_{22} -p_{12} \right) \left( p_{31} -p_{32} \right) }{ p_{11} +p_{22} -2 p_{12} } \right\} Q + \left\{ p_{33} -\frac{ \left( p_{13} -p_{23} \right)^{2} }{p_{11} +p_{22} -2 p_{12} } \right\} Q_{3} + \cdot\cdot\cdot $$

Each coefficient of $Q_{i}$ contains

$$ -\frac{ \left( p_{13} -p_{23} \right)^{2} }{p_{11} +p_{22} -2 p_{12} } $$

What I tried are as below.

$$ V_{1} = p_{11}Q_{1} + p_{12} Q_{2} + p_{13} Q_{3} + \cdot\cdot\cdot $$

$$ V_{2} = p_{21}Q_{1} + p_{22} Q_{2} + p_{23} Q_{3} + \cdot\cdot\cdot $$

Since $ V_{1} = V_{2} $ is held,

$$ p_{11}Q_{1} + p_{12} Q_{2} + p_{13} Q_{3} + \cdot\cdot\cdot= p_{21}Q_{1} + p_{22} Q_{2} + p_{23} Q_{3} + \cdot\cdot\cdot $$

$$ Q_{1} \left\{ p_{11} -p_{12} \right\} =Q_{2} \left\{ p_{22} -p_{12} \right\} + Q_{3} \left\{ p_{23} -p_{13} \right\} +\cdot\cdot\cdot $$

$$ Q_{2} \left\{ p_{12} -p_{22} \right\} = Q_{1} \left\{ p_{12} -p_{11} \right\} + Q_{3} \left\{ p_{23} -p_{13} \right\} + \cdot\cdot\cdot $$

I've been stucked from here.

Answer 1

The main thing here is that they've apparently chosen to prefer using $Q$ to using $Q_2$, so they rewrite $Q_2 = Q - Q_1$, and substitute for $Q_2$ in your calculation (FYI - \cdots will produce "$\cdots$"): $$Q_1 \left(p_{11}-p_{12}\right) =Q_2 \left(p_{22}-p_{12}\right) + Q_3 \left(p_{23}-p_{13}\right) +\cdots$$ $$Q_1 \left(p_{11}-p_{12}\right) =Q\left(p_{22}-p_{12}\right) - Q_1\left(p_{22}-p_{12}\right) + Q_3 \left(p_{23}-p_{13}\right) +\cdots$$ $$Q_1 \left([p_{11}-p_{12}] + [p_{22}-p_{12}] \right) =Q\left(p_{22}-p_{12}\right) + Q_3 \left(p_{23}-p_{13}\right) +\cdots$$ $$Q_1 \left(p_{11}+p_{22}-2p_{12}\right) =Q\left(p_{22}-p_{12}\right) + Q_3 \left(p_{23}-p_{13}\right) +\cdots$$ $$Q_1 = Q\dfrac{p_{22}-p_{12}}{p_{11}+p_{22}-2p_{12}} + Q_3 \dfrac{p_{23}-p_{13}}{p_{11}+p_{22}-2p_{12}} + Q_4 \dfrac{p_{24}-p_{14}}{p_{11}+p_{22}-2p_{12}}+\cdots$$

For the calculation of $V_3$, they just substituted this expression for $Q_1$ into $$V_3 = \left( p_{31} - p_{32} \right) Q_{1} +p_{32} Q + p_{33} Q_{3} + \cdots$$ So $$V_3 = (p_{31}-p_{32})\left(Q\dfrac{(p_{22}-p_{12})}{p_{11}+p_{22}-2p_{12}} + Q_3 \dfrac{(p_{23}-p_{13})}{p_{11}+p_{22}-2p_{12}} + \cdots\right)\\+p_{32}Q + p_{33}Q_{3} + \cdots\\ = \left(Q\dfrac{(p_{22}-p_{12})(p_{31}-p_{32})}{p_{11}+p_{22}-2p_{12}}+ Q_3 \dfrac{(p_{23}-p_{13})(p_{31}-p_{32})}{p_{11}+p_{22}-2p_{12}} + \cdots\right)\\+p_{32}Q + p_{33}Q_{3} + \cdots\\ =Q\left(p_{32}+\dfrac{(p_{22}-p_{12})(p_{31}-p_{32})}{p_{11}+p_{22}-2p_{12}}\right) + Q_3\left(p_{33}+\dfrac{(p_{23}-p_{13})(p_{31}-p_{32})}{p_{11}+p_{22}-2p_{12}}\right) + \cdots$$

The final thing they did only applies to the coefficient of $Q_3$: Noting that $p_{ij} = p_{ji}$, they had the bright idea of rewriting $$(p_{31} - p_{32}) = (p_{13} - p_{23}) = - (p_{23} - p_{13})$$ so $$(p_{23}-p_{13})(p_{31}-p_{32}) = -(p_{23}-p_{13})^2$$ and $$\left(p_{33}+\dfrac{(p_{23}-p_{13})(p_{31}-p_{32})}{p_{11}+p_{22}-2p_{12}}\right) = \left(p_{33}-\dfrac{(p_{23}-p_{13})^2}{p_{11}+p_{22}-2p_{12}}\right)$$ But again, this only affects the $Q_3$ term. It is not in the $Q$ term, and it is not in later terms either. The $Q_4$ term is $$Q_4\left(p_{34}+\dfrac{(p_{24}-p_{14})(p_{31}-p_{32})}{p_{11}+p_{22}-2p_{12}}\right)$$ (And the very fact that they used this special handling for the last term they actually show in a series you are supposed to extrapolate by pattern idicates that their idea was far less bright than they thought.)

Answer 2

I wrote in a comment that one should do a check for the equations and that may reveal a way to deduce these equation. So for the equation $(6)$ $$Q_{1} =\frac{p_{22} - p_{12} }{p_{11}+ p_{22} -2 p_{12} } Q+ \frac{\left( p_{23} -p_{13} \right) Q_{3} +\left( p_{24} -p_{14} \right) Q_{4} }{p_{11}+ p_{22} -2 p_{12} } \tag{6}$$ where I removed the $\cdots$ part, I did such a check. But I used a CAS (Maxima) to do these calculations and added some information as comments. I hope one can follow these calculations even if one is not familiar with Maxima.

(%i3) /* %i is the input line, %o is the output line
         this is equation (6), it is stored in variable e1 */
      e1:Q1 = ((p22-p12)/(p11+p22+(-2)*p12))*Q
            +((p23-p13)*Q3+(p24-p14)*Q4)/(p11+p22+(-2)*p12)
(%o3) Q1 = (Q4*(p24-p14)+Q3*(p23-p13))/(p22-2*p12+p11)
         +(Q*(p22-p12))/(p22-2*p12+p11)
(%i4) /* we replace Q by Q1 + Q2 in e1 and store the resulting equation in e2 */
      ev(e2:e1,Q = Q1+Q2)
(%o4) Q1 = (Q4*(p24-p14)+Q3*(p23-p13))/(p22-2*p12+p11)
         +((Q2+Q1)*(p22-p12))/(p22-2*p12+p11)
(%i5) /* from (2), (3) and (4) we get an equation that we store in e3: */
      e3:p11*Q1+p12*Q2+p13*Q3+p14*Q4 = p21*Q1+p22*Q2+p23*Q3+p24*Q4
(%o5) Q4*p14+Q3*p13+Q2*p12+Q1*p11 = Q4*p24+Q3*p23+Q2*p22+Q1*p21
(%i6) /* from equation e3 we can calculate Q4, we store this equation in e4 */
      e4:solve(e3,Q4)
(%o6) [Q4 = -(Q3*p23+Q2*p22+Q1*p21-Q3*p13-Q2*p12-Q1*p11)/(p24-p14)]
(%i7) /* now we check if the solution e2 is correct by inserting e4 and 
         store the resulting equation in e5 */
      ev(e5:e2,e4)
(%o7) Q1 = (Q3*(p23-p13)-Q3*p23-Q2*p22-Q1*p21+Q3*p13+Q2*p12+Q1*p11)
         /(p22-2*p12+p11)
         +((Q2+Q1)*(p22-p12))/(p22-2*p12+p11)
(%i8) /* we expand the paranthesis */
      ev(e6:e5,expand)
(%o8) Q1 = (Q1*p22)/(p22-2*p12+p11)-(Q1*p21)/(p22-2*p12+p11)
                                   -(Q1*p12)/(p22-2*p12+p11)
                                   +(Q1*p11)/(p22-2*p12+p11)
(%i9) /* bring it on the same denominator */
      e7:rat(e6)
(%o9) Q1 = (Q1*p22-Q1*p21-Q1*p12+Q1*p11)/(p22-2*p12+p11)
(%i10) /* and use the fact that p12=p21 to see that the LHS is equal to the RHS */
       ev(e8:e7,p21 = p12)
(%o10) Q1 = Q1

So following these calculations from the bottom (line %o10) to the top (line %i3) should enable you to construct a proof to deduce $(6)$.