Question 3.12: Let a1^→ = (−1, 1, 1), a2^→ = (1,−1, 1) and a3^→ = (1, 1,−1...
Let \overrightarrow{a_{1} }=\left(-1,1,1\right), \overrightarrow{a_{2} }=\left(1,-1,1\right) and \overrightarrow{a_{3} }=\left(1,1,-1\right).
(1) Try to find linear operators mapping the tetrahedron \Delta \overrightarrow{0}\overrightarrow{a_{1} }\overrightarrow{a_{2} } \overrightarrow{a_{3} } onto the tetrahedron \Delta \overrightarrow{0} \left(-\overrightarrow{a_{1} }\right) \left(-\overrightarrow{a_{2} }\right) \left(-\overrightarrow{a_{3} }\right). See Fig. 3.54(a).
(2) Try to find a linear operator mapping the tetrahedron \Delta \overrightarrow{0}\overrightarrow{a_{1} }\overrightarrow{a_{2} } \overrightarrow{a_{3} } onto the parallelogram \overrightarrow{a_{1} }\overrightarrow{a_{2} }. See Fig. 3.54(b).
(1) There are six such possible linear operators. The simplest one, say f_{1}, among them is the one that satisfies
f_{1} \left(\overrightarrow{a_{i} }\right)=-\overrightarrow{a_{i} } for 1 ≤ i ≤ 3.
In the natural basis N=\left\{\overrightarrow{e_{1} },\overrightarrow{e_{2} },\overrightarrow{e_{3} }\right\},
\left[ f_{1}\right] _{N} =\left[\begin{matrix} \overrightarrow{a_{1} } \\ \overrightarrow{a_{2} } \\ \overrightarrow{a_{3} } \end{matrix} \right] ^{-1} \left[\begin{matrix} -1 & & 0 \\ & -1 & \\ 0 & &-1 \end{matrix} \right] \left[\begin{matrix} \overrightarrow{a_{1} } \\ \overrightarrow{a_{2} } \\ \overrightarrow{a_{3} } \end{matrix} \right]=-I_{3}
\Rightarrow f_{1}\left(\overrightarrow{x} \right) =-\overrightarrow{x}=-\overrightarrow{x}I_{3}.
It is possible that \overrightarrow{a_{1} } and \overrightarrow{a_{2} } are mapped into -\overrightarrow{a_{2} } and -\overrightarrow{a_{1} } respectively while \overrightarrow{a_{3} } is to -\overrightarrow{a_{3} }. Denote by f_{2} such a linear operator. Then
f_{2}\left(\overrightarrow{a_{1} }\right) =-\overrightarrow{a_{2} },
f_{2}\left(\overrightarrow{a_{2} }\right) =-\overrightarrow{a_{1} },
f_{2}\left(\overrightarrow{a_{3} }\right) =-\overrightarrow{a_{3} }. \left(*_{1} \right)
\Rightarrow \left[ f_{2}\right] _{N} = P^{-1} \left[\begin{matrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 &-1 \end{matrix} \right] P, where P=\left[\begin{matrix} \overrightarrow{a_{1} } \\ \overrightarrow{a_{2} } \\ \overrightarrow{a_{3} } \end{matrix} \right]=\left[\begin{matrix} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 &-1 \end{matrix} \right]
= \frac{1}{2} \left[\begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} \right]\left[\begin{matrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \end{matrix} \right]\left[\begin{matrix} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 &-1 \end{matrix} \right]=\left[\begin{matrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \end{matrix} \right]
\Rightarrow f_{2}\left(\overrightarrow{x} \right) = \overrightarrow{x}\left[ f_{2}\right] _{N} =\overrightarrow{x}\left[\begin{matrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1 \end{matrix} \right]=-\overrightarrow{x}\left[\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} \right]. \left(*_{2} \right)
Notice that, \left(*_{1} \right) is equivalent to
-f_{2}\left(\overrightarrow{e}_{1} \right)+f_{2}\left(\overrightarrow{e}_{2} \right)+f_{2}\left(\overrightarrow{e}_{3} \right)=-\overrightarrow{a_{2} }, f_{2}\left(\overrightarrow{e}_{1} \right)-f_{2}\left(\overrightarrow{e}_{2} \right)+f_{2}\left(\overrightarrow{e}_{3} \right)=-\overrightarrow{a_{1} }, f_{2}\left(\overrightarrow{e}_{1} \right)+f_{2}\left(\overrightarrow{e}_{2} \right)-f_{2}\left(\overrightarrow{e}_{3} \right)=-\overrightarrow{a_{3} }
\Rightarrow f_{2}\left(\overrightarrow{e}_{1} \right)+f_{2}\left(\overrightarrow{e}_{2} \right)+f_{2}\left(\overrightarrow{e}_{3} \right)=-\left(\overrightarrow{a_{1} }+\overrightarrow{a_{2} }+\overrightarrow{a_{3} }\right)=-\left(1,1,1\right)
\Rightarrow f_{2}\left(\overrightarrow{e}_{1} \right)=-\overrightarrow{e}_{2}, f_{2}\left(\overrightarrow{e}_{2} \right)=-\overrightarrow{e}_{1} , f_{2}\left(\overrightarrow{e}_{3} \right)=-\overrightarrow{e}_{3} . \left(*_{3} \right)
This is just \left(*_{2} \right). f_{2} is diagonalizable. Similarly, both
f_{3}\left(\overrightarrow{x} \right) = -\overrightarrow{x} \left[\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} \right] and f_{4}\left(\overrightarrow{x} \right) = -\overrightarrow{x} \left[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} \right]
are another two such linear operators.
The last two linear operators are
f_{5}\left(\overrightarrow{x} \right) = -\overrightarrow{x} \left[\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} \right] and f_{6}\left(\overrightarrow{x} \right) = -\overrightarrow{x} \left[\begin{matrix} 0& 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} \right].
Both are not diagonalizable. For details, see Sec. 3.7.8.
(2) The parallelogram \overrightarrow{a_{1} }\overrightarrow{a_{2} } has the vertices at \overrightarrow{0},\overrightarrow{a_{1} },\overrightarrow{a_{1} }+\overrightarrow{a_{2} }=2 \overrightarrow{e_{3} } and \overrightarrow{a_{2} }.
Define a linear operator g: R³ → R³ as
g\left(\overrightarrow{a_{1} }\right) =\overrightarrow{a_{1} },
g\left(\overrightarrow{a_{2} }\right) =\overrightarrow{a_{2} },
g\left(\overrightarrow{a_{3} }\right) =\overrightarrow{a_{1} }+\overrightarrow{a_{2} }=2\overrightarrow{e_{3} }.
The process like \left(*_{2} \right) or \left(*_{3} \right) will lead to
g\left(\overrightarrow{x }\right)=\overrightarrow{x }\left[ g\right] _{N} =\overrightarrow{x }\left[\begin{matrix} \frac{1}{2} & -\frac{1}{2} & \frac{3}{2} \\ \\ -\frac{1}{2} & \frac{1}{2} & \frac{3}{2} \\ \\ 0 & 0 & 1 \end{matrix} \right] or
\left[ g\right] _{B} =P\left[ g\right] _{N}P^{-1}=\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{matrix} \right],
where P is as above and B=\left\{\overrightarrow{a_{1} },\overrightarrow{a_{2} },\overrightarrow{a_{3} }\right\} . g is diagonalizable and
Q\left[ g\right] _{N}Q^{-1}=\left[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} \right], where Q=\left[\begin{matrix} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -3 \end{matrix} \right].
g is a projection of R³ onto the subspace \ll \overrightarrow{a_{1} },\overrightarrow{a_{2} }\gg along \ll \left(1,1,-3\right)\gg as can be visualized in Fig. 3.54(b).
The readers are urged to find more such linear operators.
One of the main advantages of diagonalizable linear operators or matrices A is that it is easy to compute the power
A^{n}
for n ≥ 1 and n < 0 if A is invertible. More precisely, suppose
A=P^{-1}\left[\begin{matrix} \lambda _{1} & & 0 \\ &\lambda _{2} & \\ 0 & &\lambda _{3} \end{matrix} \right]P
⇒ 1. det(A) = \lambda _{1} \lambda _{2} \lambda _{3} .
2. A is invertible ⇔ \lambda _{1} \lambda _{2} \lambda _{3} \neq 0 . In this case,
A^{-1}=P^{-1}\left[\begin{matrix} \lambda^{-1} _{1} & & 0 \\ &\lambda^{-1} _{2} & \\ 0 & &\lambda^{-1} _{3} \end{matrix} \right]P.
3. Hence
A^{n}=P^{-1}\left[\begin{matrix} \lambda^{n} _{1} & & 0 \\ &\lambda^{n} _{2} & \\ 0 & &\lambda^{n} _{3} \end{matrix} \right]P.
4. tr(A) = \lambda _{1}+\lambda _{2}+\lambda _{3}.
5. For any polynomial g\left(t\right)\in P _{n}\left(R\right),
g\left(A\right)=P^{-1}\left[\begin{matrix} g\left(\lambda _{1} \right)& & 0 \\ &g\left(\lambda _{2} \right) & \\ 0 & &g\left(\lambda _{3} \right) \end{matrix} \right]P. (3.7.49)
These results still hold for any diagonalizable matrix of finite order.