In Example 11.9, assume that we start with a hybrid bred to a hybrid. Find u(1), u(2), and u(3). What would u(n) be?
\[P = \begin{array} \mbox{} & GG & Gg & gg \\ \mbox{GG} & 0.5 & 0.5 & 0 \\ \mbox{Gg} & 0.25 & 0.5 & 0.25 \\ \mbox{gg} & 0 & 0.5 & 0.5 \end{array}\]
\(u^{(n)} = uP^{n}\)
The starting vector is \((0, 1, 0)\) because we know we have a hybrid, which is Gg.
\(u^{(1)} = uP^{1}\)
\(u^{(1)} = (0, 1, 0)\begin{pmatrix} 0.5 & 0.5 & 0 \\ 0.25 & 0.5 & 0.25 \\ 0 & 0.5 & 0.5 \end{pmatrix}\)
\(u^{(1)} = ((0\times0.5+1\times0.25+0\times0), (0\times0.5+1\times0.5+0\times0.25), (0\times0+1\times0.5+0\times0.5))\)
\(u^{(1)} = ((0+0.25+0), (0+0.5+0), (0+0.5+0))\)
\(u^{(1)} = (0.25, 0.5, 0.5)\)
\(P^{2} = \begin{pmatrix} 0.5 & 0.5 & 0 \\ 0.25 & 0.5 & 0.25 \\ 0 & 0.5 & 0.5 \end{pmatrix} \times \begin{pmatrix} 0.5 & 0.5 & 0 \\ 0.25 & 0.5 & 0.25 \\ 0 & 0.5 & 0.5 \end{pmatrix}\)
\(P^{2} = \begin{pmatrix} 0.375 & 0.5 & 0.125 \\ 0.25 & 0.5 & 0.25 \\ 0 & 0.5 & 0.5 \end{pmatrix}\)
\(u^{(2)} = uP^{2}\)
\(u^{(2)} = (0.25, 0.5, 0.25)\begin{pmatrix} 0.375 & 0.5 & 0.125 \\ 0.25 & 0.5 & 0.25 \\ 0 & 0.5 & 0.5 \end{pmatrix}\)
\(u^{(2)} = ((0.25\times0.375+0.5\times0.25+0.25\times0.125),(0.25\times0.5+0.5\times0.5+0.25\times0.5),(0.25\times0.125+0.5\times0.25+0.25\times0.375))\)
\(u^{(2)} = ((0.09375+0.125+0.03125),(0.125+0.25+0.125),(0.03125+0.125+0.09375))\)
\(u^{(2)} = (0.25,0.5,0.25)\)
\(P^{3} = \begin{pmatrix} 0.5 & 0.5 & 0 \\ 0.25 & 0.5 & 0.25 \\ 0 & 0.5 & 0.5 \end{pmatrix} \times \begin{pmatrix} 0.375 & 0.5 & 0.125 \\ 0.25 & 0.5 & 0.25 \\ 0 & 0.5 & 0.5 \end{pmatrix}\)
\(P^{3} = \begin{pmatrix} 0.3125 & 0.5 & 0.1875 \\ 0.25 & 0.5 & 0.25 \\ 0.1875 & 0.5 & 0.3125 \end{pmatrix}\)
\(u^{(3)} = uP^{3}\)
\(u^{(3)} = (0.25, 0.5, 0.25)\begin{pmatrix} 0.3125 & 0.5 & 0.1875 \\ 0.25 & 0.5 & 0.25 \\ 0.1875 & 0.5 & 0.3125 \end{pmatrix}\)
\(u^{(3)} = ((0.25\times0.3125+0.5\times0.25+0.25\times0.1875),(0.25\times0.5+0.5\times0.5+0.25\times0.5),(0.25\times0.1875+0.5\times0.25+0.25\times0.3125))\)
\(u^{3} = ((0.078125+0.125+0.046875),(0.125+0.25+0.125),(0.046875+0.125+0.078125))\)
\(u^{3} = (0.25,0.5,0.25)\)
\(u^{(1)} = u^{(2)} = u^{(3)}\)
\(u^{(n)} = u^{(1)} = u^{(2)} = u^{(3)}\)
\(u^{n} = (0.25,0.5,0.25)\)