# Meaning of Eigenvalues and Eigenvectors

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?

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.

• Thank you so much! This answers it well! Oct 16, 2020 at 0:39

The two given eigenvectors span $$\Bbb R^2$$. So we can write $$\left(\begin{smallmatrix}1\\1\end{smallmatrix}\right)$$ as a linear combination of the two: $$\begin{pmatrix}1\\1\end{pmatrix} = 1.2\begin{pmatrix}0.5\\1\end{pmatrix} - 0.2\begin{pmatrix}-2\\1\end{pmatrix}$$ Now we can apply your matrix to both sides here. Using linearity on the right-hand side, the result must be $$\lambda_1\cdot1.2\begin{pmatrix}0.5\\1\end{pmatrix} - \lambda_2\cdot0.2\begin{pmatrix}-2\\1\end{pmatrix}$$ Just insert and calculate, and you get the answer.

• Oh! Thank you so much! Oct 16, 2020 at 0:39