Consider the Matrix $\pmatrix{E&t\\t&-E}$

The eigenvalues are $\lambda\_1=\sqrt{E^2+t^2}$ , $\lambda\_2=-\sqrt{E^2+t^2}$.

Consider the first (+ve) eigenvalue.

To find the eigenvectors, we write:

$$\pmatrix{E-\lambda\_1 &t\\t&-E-\lambda\_1} \pmatrix{a\\b} =\pmatrix{0\\0}$$

The problem is that there are two different ways to proceed:

$$(1)\qquad(E-\lambda\_1)a + tb = 0 \Longrightarrow b=\frac{-E+\lambda\_1}{t}$$

$$(2)\qquad ta + (-E-\lambda\_1)b = 0 \Longrightarrow a=\frac{E+\lambda\_1}{t}$$

I know the general solutions is any multiple integer of the eigenvector, but what we have above is NOT! that's something different, what is above is different eigenvectors for the same eigenvalue (not simple integer multiples...)

So what's wrong? Which one of 1) or 2) is right?