The Distance between the Eigenvalues of Hermitian Matrices

It is shown that the minmax principle of Ky Fan leads to a quick simple derivation of a recent inequality of V. S. Sunder giving a lower bound for the spectral distance between two Hermitian matrices. This brings out a striking parallel between this result and an earlier known upper bound for the spectral distance due to L. Mirsky. Let A be a Hermitian matrix of order n and let A ¿(A) denote the vector in Rn whose coordinates are the eigenvalues of A arranged as A[i](A) >••'•> A[n](A). Let A(i)(A) < ■ • ■ < A(n)(A) be the increasing rearrangement of these eigenvalues and At (A) the vector with coordinates A(j)(A), j = 1,2,..., n. The same symbols Xi [A) and Af (A) will also denote the diagonal matrices which have as their diagonal entries the components of the vectors Af [A) and Af (A), respectively. Let || • || denote any unitarily invariant norm on the space of matrices. (See [4].) This note is concerned with the following result: THEOREM. Let A and B be Hermitian matrices. Then for every unitarily invariant norm we have (1) Ul(A) Aj (ß)|| < \\A B\\ < \\Xt (A) AT(5)||. The first inequality in (1) appeared in a paper of Mirsky [4], who used a famous result of Lidskii and Wielandt to derive it. The second is proved in a recent paper of Sunder [5]. I give here another proof of the second inequality which has two attractive features: It is very short and it proceeds on exactly the same lines as the well-known proof of Lidskii, Wielandt and Mirsky for the first inequality. For illumination, I indicate how both inequalities follow from the same principle. It is an easy consequence of the minmax principle of Wielandt that for any choice 1 < ix < ■ ■ ■ < ik < n of k indices we have (2) E am (A+B) < EA w w + EA w (ß) 3 = 1 3 = 1 3 = 1 for all k — 1,2,..., n, with equality holding for k = n. (See [3, p. 242].) Writing x < y to mean that the vector x is majorised by the vector y in Rn (see [3]), we get from inequalities (2) (3) Xí(A + B)-Xi(B)<Xí(A). With a change of variables, this gives _ Xi(A)-Xl{B)<Xl(A-B). Received by the editors August 27, 1984 and, in revised form, January 14, 1985. 1980 Mathematics Subject Classification. Primary 15A42, 15A57.