How many eigenvectors have this matrix?

Question

I am trying to calculate eigenvalues and eigenvectors of this matrix

$$\frac12\begin{pmatrix} 1 & 1 & 1 & 1 \\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1 \end{pmatrix}?$$

Determinat is $\frac{1}{16}(\lambda^2-4)^2$ and it should have two eigenvalues $2$ and $-2$. I also need eigenvectors. I decided to check it by wolframalpha. However, there are 4 eigenvectors, I have only two. Where I made mistake?

Answer 1

Note that also the identity $I$ has $1$ eigenvalues but $4$ independent eigenvectors.

In your case we have 2 eigenvalues with algebraic multiplicity equal to 2 but the key point for eigenvectors is the geometric multiplicity of each eigenvalue, that is $n-r$ with $r=$rank of $(A-\lambda I)$.

In this case, if $rank(A-\lambda I)=2$ for each eigenvalue we can find $2 $ corresponding eigenvectors by the solution of $(A-\lambda I)x=0$.

Answer 2

The system $$(A-\lambda I)V =0 $$ has more than one solution for $\lambda =2$ or$\lambda =-2$ .