Pinching, Trimming, Truncating, and Averaging of Matrices

The operator norm of an n × n complex matrix A is its norm as a linear operator on the Euclidean space I C; i.e., ‖A‖ = sup{‖Ax‖ : x ∈ I C, ‖x‖ = 1}. The Frobenius norm of A is defined as ‖A‖2 = ( ∑ i,j |aij|) 1 2 = (trA∗A) 1 2 . Both these norms are used frequently in analysis of matrices. They can also be described in terms of the singular values of A − the square roots of the eigenvalues of A∗A enumerated as s1(A) ≥ · · · ≥ sn(A). We have ‖A‖ = s1(A) and ‖A‖2 = ( ∑ j s 2 j (A)) 1 2 .