Hi, I have a question on this chapter 2, pg 41 second Paragragh starting with "While any real symmetric matrix .."

Here the author says if there are two or more eigenvectors with same eigenvalue, then any set of orthogonal vectors lying in their span are also eigenvectors with same eigenvalue.

I am having trouble wraping my head around the fact that the orthogonal vectors lying in the span will have same eigenvalues.

e.g consider A = [2,1;1,2] matrix .. to solve for Av = lambda.*v we get eigenvalues lambda = 3 and 1 and eigenvectors for lambda=3 are [1,1] , [3,3] etc and for lambda =1 are [1,-1]. clearly vector [1, -1] lies in the span of both [1,1] and [3,3] and is orthogonal to both but does not share the same eigenvalue.