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 H1 otimes H2, there exist orthonormal sets { u1, ldots, un } subset H1 and { v1, ldots, vm } subset H2 such that v = sum{i =1} ^m alpha i ui otimes vi, 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 { e1, ldots, en } subset H1 and { f1, ldots, fm } subset H2. We can identify an elementary tensor ei otimes fj with the matrix ei fj ^T, where fj ^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} ei otimes fj can then be viewed as the n × m matrix ; Mv = (beta{ij}){ij} .
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GarantieWeitere Informationen
- Allgemeine Informationen
- GTIN 09786131159817
- Editor Lambert M. Surhone, Miriam T. Timpledon, Susan F. Marseken
- EAN 9786131159817
- Format Fachbuch
- Titel Schmidt Decomposition
- Herausgeber Betascript Publishing
- Anzahl Seiten 96
- Genre Mathematik
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