By Ola Bratteli
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Additional info for Derivations, Dissipations and Group Actions on C-Algebras
Then L is a lower triangular matrix with nonnegative diagonal entries and A = LL*. Suppose A is positive definite. We show the uniqueness of this factorization. Let A = L1L1 = L2L2 where L1 and L2 are lower triangular with nonnegative diagonal entries. Since A is nonsingular, the diagonal entries of L1 and L2 are nonzero and hence positive. 13) L2 1L1 = [(L2 1L1)-1]* where the matrix on the left-hand side is lower triangular while the matrix on the right-hand side is upper triangular. Thus L21 L1 is diagonal and its diagonal entries are positive.
1. Eigenvalues of Hermitian Matrices Clearly, the eigenvalues of every Hermitian matrix are real numbers. We always denote the eigenvalues of A E Hn in decreasing order by A1(A) > A2 (A) > ... > An (A). 1 (The Min-Max Expression, Courant-Fischer). Let A E Hn, 1 < k < n. Then Ak(A) = max min x*Ax = SCC xES dim S=k 114=1 where S is a subspace of Cn and 11 min SCC max x*Ax, xES dim S=n-k+l 114=1 11 is the Euclidean norm. Proof. Denote A3 = A3 (A), j = 1, ... , n. Let S be any given subspace of Cn with dim S = k.
Is I ji,. . , jt] the (m - s) x t submatrix of A and columns ji, . , ii,... . obtained from A [1, ... , m I ji,.. , jt] by deleting its rows i 1, ... , is. ,ik) Ii <_ it < ... < ik < n}. 4. Compound Matrices 47 For clarity we define only the compound matrices of square matrices. This is the case we need in the book. The compound matrices of rectangular matrices can be similarly defined. ]. Arrange the elements of I' (k, n) in the lexicographical order as al, a2 ... , a(n). For example, the elements of I'(2, 4) are arranged as k (1, 2), (1, 3), (1, 4), (2) 3), (2) 4), (3, 4).
Derivations, Dissipations and Group Actions on C-Algebras by Ola Bratteli