Question 2.1: The linear operator A = [λ o o o] with λ ≠ 0 has the pr...
The linear operator A = \begin{bmatrix} \lambda & 0 \\ 0 & 0 \end{bmatrix} with λ ≠ 0
has the properties:
1. \overrightarrow{e_{1} }A = λ\overrightarrow{e_{1} } and \overrightarrow{e_{2} } A = \overrightarrow{0} =0\overrightarrow{e_{2} } . λ and 0 are called eigenvalues of A
with corresponding eigenvectors \overrightarrow{e_{1} } and \overrightarrow{e_{2} }, respectively.
2. Ker(A) = «\overrightarrow{e_{2} }», while Im(A) =«\overrightarrow{e_{2} }» is an invariant line (subspace)
of A, i.e. A maps \overrightarrow{e_{1} } into it self.
3. A maps every line \overrightarrow{x_{0} } +«\overrightarrow{x}», where \overrightarrow{x } is linearly independent of \overrightarrow{e_{2} },one-to-one and onto the line «\overrightarrow{e_{1} }»
See Fig. 2.43(a) and (b). In Fig. 2.43(b), we put the two plane coincide
and the arrow signs indicate how A preserves ratios of signed lengths of
segments.
Operators
\begin{bmatrix} \lambda & 0 \\ 0 & 0 \end{bmatrix} , \begin{bmatrix} 0 & 0 \\ \lambda & 0 \end{bmatrix} and \begin{bmatrix} 0 & 0 \\ 0 & \lambda \end{bmatrix} where λ ≠ 0
are all of this type.
