Question 2.9: The linear operator A=[0 1 b a], where ab ≠ 0 has the follow...
The linear operator
A=\begin{bmatrix} 0 & 1 \\ b & a \end{bmatrix} , where ab ≠ 0
has the following properties:
(1) a² + 4b > 0. Let \lambda_{1}=\frac{a+\sqrt{a^{2}+4b} }{2} and \lambda_{2}=\frac{a-\sqrt{a^{2}+4b} }{2} and \overrightarrow{v_{i}} =\left(b,\lambda_{i}\right) for i = 1, 2.
- \overrightarrow{v_{i}} A=\lambda_{i}\overrightarrow{v_{i}} , i = 1, 2. Thus, \lambda_{1} and \lambda_{2} are eigenvalues of A with corresponding eigenvectors \overrightarrow{v_{1}} and \overrightarrow{v_{2}}.
- \ll \overrightarrow{v_{1}} \gg and \ll \overrightarrow{v_{2}} \gg are invariant lines (subspaces) of R² under A.
In the basis B = \left\{\overrightarrow{v_{1}},\overrightarrow{v_{2}}\right\}, A can be represented as
\left[A\right]_{B}=PAP^{-1}= \begin{bmatrix} \lambda_{1} & 0 \\ 0 & \lambda_{2} \end{bmatrix} , where P =\left[\begin{matrix} \overrightarrow{v_{1}} \\ \overrightarrow{v_{2}} \end{matrix} \right]=\begin{bmatrix} b & \lambda_{1} \\ b & \lambda_{2} \end{bmatrix}.
(see Exs. <B> 4, 5 of Sec. 2.4 and Sec. 2.7.3.) See Fig. 2.53.
(2) a² + 4b = 0. Let λ = \frac{a}{2} ,\frac{a}{2} and \overrightarrow{v}=\left(-a,2\right).
- \overrightarrow{v} A= \lambda \overrightarrow{v}.λ is an eigenvalue of multiplicity 2 of A with corresponding eigenvector \overrightarrow{v}.
- \ll \overrightarrow{v} \gg is an invariant line (subspace) of R² under A.
In the basis B =\left\{\overrightarrow{v},\overrightarrow{e_{1}}\right\}, A can be represented as
\left[A\right]_{B}=PAP^{-1}= \begin{bmatrix} \lambda & 0 \\ \frac{1}{2} & \lambda \end{bmatrix}= \lambda \begin{bmatrix} 1 & 0 \\ \frac{1}{2\lambda} & 1 \end{bmatrix}=\lambda I_{2}+\begin{bmatrix} 0 & 0 \\ \frac{1}{2} & 0 \end{bmatrix},
where P =\left[\begin{matrix} \overrightarrow{v} \\ \overrightarrow{e_{1}} \end{matrix} \right]=\begin{bmatrix} -a & 2\\ 1 & 0 \end{bmatrix}.
This is the composite map of a shearing followed by an enlargement with scale λ. See Figs. 2.52 and 2.54.
(3) a² + 4b < 0. A does not have any invariant line. Notice that
\begin{bmatrix} 0 & 1 \\ b & a \end{bmatrix}=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} b & 0 \\ 0 & 1 \end{bmatrix}+\begin{bmatrix} 0 & 0 \\ 0 & a \end{bmatrix}.
Hence, the geometric mapping property of A can be described as follows in N =\left\{\overrightarrow{e_{1}},\overrightarrow{e_{2}}\right\}.
See Fig. 2.55.
The operator
\begin{bmatrix} a & 1 \\ b & 0 \end{bmatrix}, where ab ≠ 0
is of the same type.
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