title: conjugate gradient with a positive semidefinite matrix
body: <p>I'm working on conjugate gradient to solve Ax=b when A is symmetric and positive semidefinite.</p>
  
  <p>When A is symmetric and positive semidefinite, is (A+λ I), where λ is positive and I is an identity matrix, always positive definite? Then can we use (A+λ I) instead of A in CG since (A+λ I) is symmetric and positive definite?</p>
  
  <p>When A is positive semidefinite with many repeated eigenvalues of zeros, are both A and (A+λ I) not full rank? How does CG behave when the matrix is not full rank?</p>
  
  <p>Many thanks!</p>