A special linear operator on

Abstract

For any n×n matrix A in the space its Ky Fan K-norm ∣⋅∣ k is defined by the sum of its k largest singular values. It is known that a linear operator T on is an isomeiry for ∣⋅∣ k if there exist orthogonal matrices U and V such that or However, in the unique case of (n,k)=(4,2), a linear isometry for ∣⋅∣2 can also be given by the composition of the above form with a linear operator L possessing many remarkable properties. For example, let R↺+(↺−) denote the set of all 4×4 orthogonal matrices A with det(A)=1(det(A)=−1), and let R denote the set of all matrices in with singular values 2,0,0,0. Then L(R)=↺+,L(↺+)=and let R and denote the set of all matrices in M 4(R) with singular values 2,0,0,0,. Then L(R)=↺+,L(↺+)= R and L(↺−)=↺−. Moreover the singular values and the determinant of L(A) can be easily determined by those of the matrix A. There now exist two proofs for these interesting properties, both of which rely heavily on the theory of real quaternions. The purpose of this paper is to give eiemeniary co...