CSC 576: Linear System

• σi(A) denotes the ith largest singular value of A, σmin(A) denotes the minimal nonzero singular value of A and σmax(A) denotes the maximal singular value of A; • If U has orthogonal columns, that is, UTU = I, then ‖Ux‖ = ‖x‖ and ‖U>y‖ ≤ ‖y‖; • ‖Ax‖ ≤ σmax(A)‖x‖, where σmax(A) denotes the largest singular value of A; • Note that we do NOT have in general ‖Ax‖ ≥ σmin(A)‖x‖, where σmin(A)(> 0) denotes the minimal nonzero singular value of ‖A‖. However, we have ‖AATx‖ ≥ σmin(A)‖Ax‖ (due to ‖AATx‖ = ‖UΣ2UT = ‖Σ2UTx‖ ≥ σmin(Σ)‖ΣUx‖ = σmin(Σ)‖V ΣUTx‖ = σmin(A)‖Ax‖); • The compact SVD of A is A = UΣV T . We have span(A) = span(U) and span(AT ) = span(V );