Given a symmetric, positive semi-definite matrix $M$ with $p$ dimensions, and its eigenvalues are $\lambda_1=\lambda_2>…>\lambda_p$, how to show that the corresponding eigenvectors $u_1$ and $u_2$ are not unique and span 2-dimensional subspace?
Tried:
let $U = (u_1, u_2,…,u_p)$, $D = $diag$(\lambda_1,…,\lambda_p)$, then we have $M = UDU^\intercal$, and due to orthogonality of $U$, we have $D = U^\intercal MU$, and satisfies $u_1^\intercal Mu_1=u_2^\intercal Mu_2$.
Also, we know that $u_1^\intercal u_1=u_2^\intercal u_2 = 0$, as well as $u_1^\intercal u_1 = u_2^\intercal u_2 =0$.
I just don't know how to make use of these relations…
Any ideas??
Best Answer
You know that $M=U D U^T$, with $D$ diagonal and $U$ unitary. Suppose $D$ is ordered so that $D= \operatorname{diag}(\lambda_1,\lambda_1,\lambda_3,...,\lambda_p)$.
Then $(M-\lambda_1 I)x = 0$ iff $(D-\lambda_1 I) U^T x = 0$.
Note that $D-\lambda_1 = \operatorname{diag}(0,0,\lambda_3-\lambda_1,...,\lambda_p-\lambda_1)$ and let $y_1 = U e_1, y_2 = U e_2$, then $y_1,y_2$ are linearly independent and $(D-\lambda_1 I) U^T y_k = (D-\lambda_1 I) e_k = 0$ (for $k=1,2$).
Hence $\operatorname{sp} \{ y_1,y_2\} \subset \ker (M-\lambda_1 I)$.
Since $\operatorname{sp} \{ y_1,y_2\} = \operatorname{sp} \{ y_1-y_2, y_1+y_2 \}$, we see that the eigenvectors are not 'unique'.
As an aside, eigenvectors are never unique. Even if normalized, if $u$ is an eigenvector, so is $-u$.