Compound matrices and some classical inequalities Tin-Yau Tam Mathematics & Statistics Auburn University Dec. 3, 04
We discuss some elegant proofs of several classical inequalities of matrices by using compound matrices: Weyl, Kostant, Yamamoto, Schur
I. Compound matrices A C n n, 1 k n. The kth compound of A is an ( ( ) n k) n k complex matrix Ck (A) whose elements are where α, β Q k,n, and C k (A) α,β = det A[α β], (1) Q k,n = {ω = (ω(1),..., ω(k)) : 1 ω(1) < < ω(k) n} Example: (1) k = 1, C 1 (A) = A, (2) k = n, C n (A) = det A. Example: n = 3 and k = 2 C 2 (A) = det A[1, 2 1, 2] det A[1, 2 1, 3] det A[1, 2 2, 3] det A[1, 3 1, 2] det A[1, 3 1, 3] det A[1, 3 2, 3] det A[2, 3 1, 2] det A[2, 3 1, 3] det A[2, 3 2, 3]
C k : GL n (C) GL ( n k )(C) is a group representation: C k (AB) = C k (A)C k (B) (Cauchy-Binet Formula) C k (I n ) = I ( n k ), (2) C k(a 1 ) = [C k (A)] 1 λ 1,..., λ n = the e-values of A in nonincreasing order λ 1 λ n The singular values of A are the nonnegative square roots of the e-values of the p.s.d. A A, also in nonincreasing order: α 1 α n. Remark: α 1 = max x 2 =1 Ax 2
Proposition 1: Let A C n n. 1. C k (A ) = [C k (A)]. Thus C k maps unitary matrices to unitary matrices. 2. The e-values of C k (A) are λ ω(j), ω Q k,n 3. The singular values of C k (A) are α ω(j), ω Q k,n 4. If A is upper triangular, so is C k (A) and the diagonal entries are a ω(j),ω(j), ω Q k,n
The kth additive compound of A: k (A) = d dt t=0 C k (I + ta), (2) or equivalently, C k (I + ta) = I + t k (A) + t 2 R + Example: If n = 3 and k = 2, then k (A) = a 11 + a 22 a 23 a 13 a 32 a 11 + a 33 a 12 a 31 a 21 a 22 + a 33. In general 1 (A) = A and n (A) = tr A. k : C n n C ( n k ) ( n k ) is an algebra representation of the Lie algebras equipped with the bracket operation [A, B] = AB BA. k = derivative of C k : GL n (C) GL ( n k )(C) at the identity.
Proposition 2: Let A C n n. 1. k (A ) = k (A). If A is Hermitian, so is k (A). 2. The e-values of k (A) are k λ ω(j), ω Q k,n. 3. The diagonal entries of k (A) are k a ω(j),ω(j), ω Q k,n
II. Weyl s Inequalities Weyl: Let A C n n. Then n λ j λ j = n α j, k = 1,..., n 1, (3) α j. (4) Proof: If x C n is a unit e-vector of A associated with λ 1, then Thus λ 1 = λ 1 x 2 = Ax 2 α 1. λ 1 α 1. (5) Recall α 1 α n and λ 1 λ n.
The largest singular value of C k (A) is α j The eigenvalue of maximum modulus of C k (A) is λ j Apply (5) on C k (A) and (3) follows. By considering the determinant of A, we have (4). Remark: Converse is true and is due to Horn. A generalization is true for real semisimple noncompact Lie groups (complete multiplicative Jordan decomposition).
III. Kostant s nonlinear convexity theorem A GL n (C), QR decomposition asserts: A = QR, Q U(n), R is upper with positive diagonal entries and the decomposition is unique. QR = Gram-Schmidt process Let a 1 a n > 0 (after rearrangement) be the diagonal entries of R. Each a i = distance between a column of A and the subspace spanned by the previous columns.
Kostant: Let A GL n (C). Then a j n a j = n α j, k = 1,..., n 1, (6) α j. (7) Proof: Suppose a 1 = the ith diagonal entry of R By the Gram-Schmidt process a 1 A i 2 where A i is the ith column of A. So a 1 α 1. (8) Notice that C k (A) = C k (Q)C k (R) is the QR-decomposition of C k (A)
The largest diagonal entry of C k (R) is a j. The largest singular value of C k (A) is k α j. Apply (8) to C k (A) to have (6). Determinant consideration leads to (7). Remark: The converse is true. The general result is about Iawasawa decomposition G = KAN of a noncompact semisimple Lie group G. Another proof: Q 1 A = R. Apply Weyl s Theorem to
IV. Yamamoto s theorem Classical result: lim m Am 1/m 2 = λ 1, (9) where A 2 = α 1 is the spectral norm of A. Yamamoto: Let A C n n. Then lim m [α i(a m )] 1/m = λ i, i = 1,..., n. (10) Proof: Recall α 1 α n, λ 1 λ n.
Apply (9) on the compound matrix C k (A): lim m [α 1(A m )] 1/m [α k (A m )] 1/m = lim m [α 1(A m ) α k (A m )] 1/m = lim m [α 1(C k (A m )] 1/m = λ 1 (C k (A)) = λ j, k = 1,..., n 1. Let 1 r n such that λ r 0 but λ r+1 = 0. From the above computation, lim m [α j(a m )] 1/m = λ j = 0, j = 1,..., r, and lim m [α j(a m )] 1/m = λ j = 0, j = r + 1,..., n. Remark: Yamamoto s theorem is very recently extended to noncompact Lie groups (KA + K decomposition).
V. Schur s inequalities Schur: Let A C n n be Hermitian with diagonal entries d 1 d n (after rearrangement) and e-values λ 1 λ n. Then k d j n d j = k n λ j, k = 1,..., n 1, (11) λ j. (12) Proof: By the spectral theorem for Hermitian matrices, there exists a unitary matrix U such that A = Udiag (λ 1,..., λ n )U. The diagonal of A is ( u ij 2 )λ, λ = (λ 1,..., λ n ) T.
The matrix ( u ij 2 ) is doubly stochastic, d 1 λ 1. (13) Applying (13) on k (A) and by Proposition 2, we get (11). By considering the trace of A, we have (12). Remark: The converse is true and is due to Horn. A generalization is true for any semisimple Lie algebra of a noncompact Lie group (Cartan decomposition and adjoint orbit). Another proof: Use interlacing inequalities for a Hermitian matrix and its principal submatrices.