Understanding eigen vectors

Question

I have a matrix, $A=\begin{pmatrix}2&1\\ 1&2\end{pmatrix}$, when I compute for the eigenvalues ($\lambda_i$) and eigenvectors ($v_i$) I get $v_1=\begin{bmatrix}1\\-1\end{bmatrix}$ for $\lambda_1=1$ and $v_2=\begin{bmatrix}1\\1\end{bmatrix}$ for $\lambda_2=3$. Now, I know $Av_i=\lambda v_i$; but what is the importance of these values? I have plotted all these values on a graph but for me they don't make any sense. Can someone explain?

Answer 1

They will be pretty useful when you want to diagonalize your matrix, that is to say find a matrix $P$ so that $\det P \ne 0$ and a diagonal matrix $D$, with :

$$ D = P^{-1} A P $$

The matrix $P$ is here given by your two eigenvectors and the matrix $D$ has your eigenvalues as diagonal coefficients.

For example, try to compute $A^n$ for $n$ large, you will see that $(P D P^{-1})^n = P D^n P^{-1} = A^n$ is much faster, especially when A itself is big. Furthermore, $A$ and $D$ have many common properties (same rank, same trace, ...)

I advise you to take a look at http://mathworld.wolfram.com/MatrixDiagonalization.html or https://www.quora.com/Linear-Algebra-Why-is-matrix-diagonalization-important-and-what-is-used-for and many other websites that can guide you and explain it in a way you might find more understandable.

You also may want to take a look at I need an intuitive explanation of eigenvalues and eigenvectors for a more detailed answer

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In your example, you had

$$ v_1=\begin{bmatrix}1\\-1\end{bmatrix} ~~\text{for}~~ \lambda_1=1\\ v_2=\begin{bmatrix}1\\1\end{bmatrix} ~~\text{for}~~ \lambda_2=3 $$

This gives you

$$ P = \begin{bmatrix}1&1\\-1&1\end{bmatrix} ~~\text{and}~~ D = \begin{bmatrix}1&0\\0&3\end{bmatrix} $$

And we have

$$ P^{-1} A P = D ~~\text{ that is to say}~~ A = P D P^{-1} $$