1.3. INEQUALITIES ON SINGULAR VALUES, I 3 The iin(A) are called the singular values of A. (1.2) is called the canonical expansion for A. 1.3. Inequalities on Singular Values, I The basis of the inequalities on singular values we discuss now is the elementary equality AXn(A) = /in(A*) (1.3) (which follows from (1.2), or alternatively from the fact that AA* and A*A have the same non-zero eigenvalues with the same multiplicities [87]) and the following: THEOREM 1.5. fin(A) = min [ max 11^4^11] 4i,...,cf)n-i if€[fii...,4n-i]± U\\=i Theorem 1.5 follows from the min-max characterization (see [254, Sec- tion XIII.1]) of the eigenvalues of -A*A if we note that ||A^||2 = (ip,A*Atj ) and that — fin(A)2 is the n-th eigenvalue of —A*A counting from the bottom. THEOREM 1.6. For any compact A and bounded B, fin(AB) \\B\\fjLn(A) (1.4a) Hn(BA) \\B\\nn(A) (1.4b) PROOF. (1.4a) follows from (1.4b) and (1.3) since (1.3) implies that /xn(AB) = /in(£*A*) ||B*||/in(A*) = \\B\\fin(A). To prove (1.4b), we need only use Theo- rem 1.5 and \\BAi/\\ \\B\\ \\A^\\. D REMARK AND WARNING. AB and BA have the same non-zero eigenvalues but they may not have the same singular values: For example, if A= (%Q), B = (Q§), then AB = 0, but BA = A, so /ii(AB) - //2(AB) = 0 but fii(BA) = 1, n2(BA) = 0. Theorem 1.6 is the first (take m = 0) of a family of inequalities of Fan [112] which read for n, ra 0, /x n + m + i(AB) /xn+i(A)/im+i(B) We will not need these, but the following inequalities of Fan [112] will be useful: THEOREM 1.7 (Fan [112]). Let A and B be compact. Then forn0,m 0, lin+rn+1{A + B) /in+i(A) + (B) (1.5) PROOF. Let Q„(^i,.. .,/„ A) = max{\\A(p\\ | ||^|| = 1, V € [tfi,...,^]- 1 -}. Since \\(A + B)i\\\\AiP\\ + \\Bi\\, (0i, ...,4 n+mi B) Qn(4i,---,4n A) + Qm(4n+1,...,4n+m B) Minimizing over pi,..., 4n+m and using Theorem 1.5, Fan's inequality (1.5) results. •

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