Let v be the k by k matrix whose first n columns are those

Question

Question: Show that the nonzero eigenvalues of A and B are the same.

Let V be the K by K matrix whose first N columns are those of the matrix C^†UL^-1/2 and whose remaining K - N columns are any mutually orthogonal norm-one vectors that are all orthogonal to each of the first N columns. Let M be the N by K matrix with diagonal entries M_nn = √λ_n for n = 1, ..., N and whose remaining entries are zero. The nonzero entries of M, √λ_n are called the singular values of C. The singular value decomposition (SVD) of C is C = UMV^†. The SVD of C^† is C^† = V M^TU^†.