Schmidt Decomposition
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High Quality Content by WIKIPEDIA articles! Let H1 and H2 be Hilbert spaces of dimensions n and m respectively. Assume n geq m. For any vector v in the tensor product H_1 otimes H_2, there exist orthonormal sets { u_1, ldots, u_n } subset H_1 and { v_1, ldots, v_m } subset H_2 such that v = sum_{i =1} ^m alpha _i u_i otimes v_i, where the scalars i are non-negative and, as a set, uniquely determined by v. [edit] Proof The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases { e_1, ldots, e_n } subset H_1 and { f_1,...
High Quality Content by WIKIPEDIA articles! Let H1 and H2 be Hilbert spaces of dimensions n and m respectively. Assume n geq m. For any vector v in the tensor product H_1 otimes H_2, there exist orthonormal sets { u_1, ldots, u_n } subset H_1 and { v_1, ldots, v_m } subset H_2 such that v = sum_{i =1} ^m alpha _i u_i otimes v_i, where the scalars i are non-negative and, as a set, uniquely determined by v. [edit] Proof The Schmidt decomposition is essentially a restatement of the singular value decomposition in a different context. Fix orthonormal bases { e_1, ldots, e_n } subset H_1 and { f_1, ldots, f_m } subset H_2. We can identify an elementary tensor e_i otimes f_j with the matrix e_i f_j ^T, where f_j ^T is the transpose of fj. A general element of the tensor product v = sum _{1 leq i leq n, 1 leq j leq m} beta _{ij} e_i otimes f_j can then be viewed as the n × m matrix ; M_v = (beta_{ij})_{ij} .
