I was reading about eigenvalues and eigenvectors.
Consider an eigenvalue $ \lambda $ of a geometric multiplicity of 2.
If the geometric multiplicity of $ \lambda $ is 2, therefore, we can write the eigenvector $ x $ as a sum of two independent vectors say $$ x=a+b $$
If I am thinking in the correct direction these vectors should also give the same eigenvalue of $\lambda$ for $A$. Therefore putting the value of $ x $ in the first equation we get
$$ A(a+b)= \lambda(a+b) $$ $$ Aa+Ab= \lambda a +\lambda b $$
I am stuck here. How can I prove that $ a $ and $ b $ have the same eigenvalue $ \lambda $?
A similar example from the book I am reading
Here also we get an eigenvector corresponding to $ F $ as a linear combination of two linearly independent vectors $ e1 $ and $ e2 $. I just wanna confirm whether these linear independent vectors by themself give the same eigenvalue $\lambda$.