eigenvector transformation

Question

Say you have the following matrix A in $R^2 \rightarrow R^2$:

$ \begin{bmatrix} 7 & -10 \\ 5 & -8 \end{bmatrix} $

Thus the eigenvalues/eigenvectors are: 2 $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ and -3 $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$.

Thus the eigenspace matrix is $\begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}$.

Say you have the vector v(x,y) of (2,3), thus Ax = [-16, -14].

I'm confused as to how does the eigenspace and eigenvalues allow me to easily see what A is doing to the vector (2,3)?

How do I apply the eigenvalues/eigenspace on vector v(2,3) to see what A is doing to it?

Answer 1

You need to write $(2,3)$ as a linear combination of the eigenvectors.

In this case, $(2,3) = -(2,1) + 4 (1,1)$, so $$A \begin{bmatrix}2 \\ 3 \end{bmatrix} = - A \begin{bmatrix}2 \\ 1 \end{bmatrix} + 4 A \begin{bmatrix}1\\ 1 \end{bmatrix}$$ and then you can use the fact that these are eigenvectors to easily complete the computation and understand how the "stretching" along the two "eigen-directions" describes the action of $A$ on $(2,3)$.

More generally, you should read about diagonalizing $A$.

Answer 2

Keep in mind that $A(2,1) = 2(2,1)$ and $A(1,1) = -3(1,1)$ by def of eigenvalues and eigenvectors. Can you write the desired vector $(2,3)$ as a linear combination of $(2,1)$ and $(1,1)$? Note that $-1(2,1)+4(1,1) = (2,3)$.

Hence, you can simplify the calculation to $$A(2,3) = -1*2*(2,1)+4*(-3)*(1,1) = (-4,-2)-(12,12) = (-16,-14).$$