Kernel of orthogonal projection on an eigenspace

Question

let $Q$ be a $d\times d $-matrix and $P:\mathbb{R}^d \to \mathbb{R}^d$ be the orthogonal projection on the eigenspace $E_0 $ of $Q$. Why is the kernel of the projection the sum of the other eigenspaces of $Q$? Is it right at all?

Thanks, Thomas

Answer 1

No, it is not right. Suppose that $d=2$, and that $Q=\left[\begin{smallmatrix}0&1\\0&1\end{smallmatrix}\right]$. Then $E_0=\langle(1,0)\rangle$ and the kernel of the orthogonal projection on that space is $\langle(0,1)\rangle$. However, the direct sum of the other eigenspaces is $\langle(1,1)\rangle$.