Given Information
We are given with following dynamical system \[ \mathbf { x } _ { k + 1 } = A \mathbf { x } _ { k } , \text { where } A = \left[ \begin{array} { c c } { 1.7 } & { - .3 } \\ { - 1.2 } & { .8 } \end{array} \right] \]

Step-1: Eigenvalues of A:
Compute the characteristic polynomial: \[ \begin{aligned} \operatorname { det } ( A - \lambda I ) & = \left| \begin{array} { c c } { 1.7 - \lambda } & { - 0.3 } \\ { - 1.2 } & { 0.8 - \lambda } \end{array} \right| \\ & = ( 1.7 - \lambda ) ( 0.8 - \lambda ) - 0.36 \\ & = \lambda ^ { 2 } - 2.5 \lambda + 1 \end{aligned} \] The eigenvalues are: \[ \begin{aligned} \lambda & = \dfrac { 2.5 \pm \sqrt { ( - 2.5 ) ^ { 2 } - 4 ( 1 ) } } { 2 } \\ & = \dfrac { 2.5 \pm \sqrt { 2.25 } } { 2 } \\ & = \dfrac { 2.5 \pm 1.5 } { 2 } \\ & = 2 \text { and } 0.5 \end{aligned} \]

Step-2:
Since one of the Eigenvalue 2 is greater than 1 and an Eigenvalues is less than 1, therefore origin is a saddle point for the dynamical system.

To find the direction of greatest attraction, we need to find the Eigenvector corresponding to Eigenvalue less than 1 magnitude.

To find the direction of greatest repulsion, we need to find the Eigenvector corresponding to Eigenvalue more than 1 magnitude.

Step-3:
Find the eigenvectors for the eigenvalues 2.0

\[ \begin{array} { r } { [ A - ( 2.0 ) I ] \mathbf { v } _ { 1 } = \mathbf { 0 } } \\ { \left[ \begin{array} { r r r } { 1.7 - 2.0 } & { - 0.3 } \\ { - 1.2 } & { 0.8 - 2.0 } \end{array} \right] \left[ \begin{array} { l } { x _ { 1 } } \\ { x _ { 2 } } \end{array} \right] = 0 } \\ { \left[ \begin{array} { r r r } { - 0.3 } & { - 0.3 } \\ { - 1.2 } & { - 1.2 } \end{array} \right] \left[ \begin{array} { l } { x _ { 1 } } \\ { x _ { 2 } } \end{array} \right] = 0 } \end{array} \] The system of equations \[\begin{array}{l} - 0.3{x_1} - 0.3{x_2} = 0\\ - 1.2{x_1} - 1.2{x_2} = 0 \end{array}\] Therefore, the eigenvector is:

\[{v_1} = \left[ {\begin{array}{*{20}{c}} { - 1}\\ 1 \end{array}} \right]\]



Step-4:
Find the eigenvectors for the eigenvalues 0.5

\[ \begin{array} { r } { [ A - ( 0.5 ) I ] \mathbf { v } _ { 2 } = \mathbf { 0 } } \\ { \left[ \begin{array} { r r } { 1.7 - 0.5 } & { - 0.3 } \\ { - 1.2 } & { 0.8 - 0.5 } \end{array} \right] \left[ \begin{array} { l } { y _ { 1 } } \\ { y _ { 2 } } \end{array} \right] = 0 } \\ { \left[ \begin{array} { r r r } { 1.2 } & { - 0.3 } \\ { - 1.2 } & { 0.3 } \end{array} \right] \left[ \begin{array} { l } { y _ { 1 } } \\ { y _ { 2 } } \end{array} \right] = 0 } \end{array} \] The system of equations \[\begin{array}{l} 1.2{y_1} - 0.3{y_2} = 0\\ - 1.2{y_1} + 0.3{y_2} = 0 \end{array}\] Therefore, the eigenvector is:

\[{v_2} = \left[ {\begin{array}{*{20}{c}} 1\\ 4 \end{array}} \right]\]



Step-5: The directions
The greatest attraction is through the origin and the Eigenvector $v_2$ and

then the greatest repulsion is through the origin and the Eigenvector $v_1$ and