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In my workings to find the eigenvector of an eigenvalue, I run into some issues. I've always relied on Mathematica to compute the eigenvectors and while I can perform the calculation by hands, it seems, the equation I have are sometimes 'gibberish'.

$A=\begin{bmatrix} -2 &0 \\ 0&2 \end{bmatrix}$

The eigenvalues are -2,2

To solve for the eigenvector:

$A\vec{v}=\lambda \vec{v}$

$\begin{bmatrix} -2 &0 \\ 0&2 \end{bmatrix}$$\begin{bmatrix} v_{1}\\ v_{2} \end{bmatrix}=-2\begin{bmatrix} v_{1}\\ v_{2} \end{bmatrix}$

The equation is

$-2v_{1}=-2v_{1}$

$2v_{2}=-2v_{2}$ which is gibberish.

Any help is appreciated.

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The second equation is true if and only if $v_2=0$, the first equation is automatically true. So the solutions are all vectors with $v_2=0$.

Something like this will always happen if you correctly selected your eigenvalue; at least one equation will always be redundant to the others.

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Note that, for a square matrix $A$, an eigenvector $v$ and its corresponding eigenvalue $\lambda$ satisfy:

$Av=\lambda v$

or equivalently,

$(A-\lambda I)v = 0$.

This means that any vector in the null space of $A-\lambda I$ is an eigenvector, so your system of equations will always be underdetermined.

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