Let's say we have some transformation matrix:
$$\begin{pmatrix} 2 & 2\\ 2 & 5 \end{pmatrix}$$
The eigen vectors are:
$$\lambda_1 = 6 \quad \lambda_2 = 1$$
With eigenvectors:
$$\begin{pmatrix} 0.5 \\ 1 \end{pmatrix} \quad \begin{pmatrix} -2 \\ 1 \end{pmatrix}$$
And we know that, if we multiply this matrix by a (1,1) vector, we get:
$$\begin{pmatrix} 2 & 2\\ 2 & 5 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 4 \\ 7 \end{pmatrix}$$
I have the question of "what do the eigenvalues and eigenvectors tell us about the transformation matrix". I know that the eigenvalue means scale and eigenvector means direction, but how do I get the (4,7) if I JUST know the eigenvalues and eigenvectors?
Best Answer
If you only know the eigenvalues and the eigenvectors, then you can write ${1\choose 1}$ as a linear combination of these eigenvectors, and then you can use the fact that the given matrix acts as a linear operator on these vectors, and so you can apply the appropriate scalar to these vectors.
You need to solve $a\begin{pmatrix} 0.5 \\ 1 \end{pmatrix} + b\begin{pmatrix} -2 \\ 1 \end{pmatrix}=\begin{pmatrix} 1 \\ 1 \end{pmatrix}$, which is equivalent to
$$\begin{pmatrix} 0.5 & -2\\ 1 & 1 \end{pmatrix}\begin{pmatrix} a \\ b \end{pmatrix}=\begin{pmatrix} 1 \\ 1 \end{pmatrix}.$$
So you can apply the inverse of this matrix to both sides and figure out what $a$ and $b$ are.