Suppose $A_{n\times n}$ is a symmetric and positive semidefinite matrix, and Rank(A)=k. I know that $A$ has k nonzero eigenvalues and corresponding orthogonal eigenvectors $v_1,\ldots,v_k$. I have two questions:
(1) I wonder whether eigenvectors corresponding to remaining zero eigenvalues is orthogonal to $v_1,\ldots,v_k$.
(2) I wonder whether eigenvectors corresponding to remaining zero eigenvalues is distinct or same.
Best Answer
The eigenvectors corresponding to the zero eigenvalues form a basis for $\mathcal{N}(A)$. Now, this basis is independent of the basis for $R(A)\setminus\mathcal{N}(A)$ which can be easily verified. Thus, this two sets of vectors can together form a basis for $R(A)$ and from that using Gram-Schmidt orthogonalization, one can form an orthogonal set of vectors.