Linear Algebra And Its Applications by Lay D.C.

By Lay D.C.

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Vnk . Here Vn is a linear space of dimension n. An operator acting on one V , can be raised to entire Fock space by a trivial comultiplication: V −→ . . + . . ⊗ 0 ⊗ V ⊗ 0 ⊗ 0 ⊗ . . + . . ⊗ 0 ⊗ 0 ⊗ V ⊗ 0 ⊗ . . + . . – the sum is over all possible positions of V . Similarly, an operator acting on a pair V1 ⊗ V2 us raised to an infinite double sum and so on. In obvious way an exponential raise to the Fock space exp⊗ is defined. For a scalar λ exp⊗ (λ) = eλ . . ⊗ I ⊗ I ⊗ . . , where I is the unit operator.

5). Its degree dn|r ≡ degS Dn|r (S) is typically smaller than degn|r ≡ degT Dn|r (T ). We call hypercubic symmetric tensors of the form n ai δii1 . . ir = i=1 diagonal. They are direct generalizations of diagonal matrices. Discriminant of diagonal tensor is a power of (a1 . . an ), the power depends on what is considered: the r-linear discriminant Dn|r = (a1 . . an )degn|r /n or the symmetric r-form one Dn|r = (a1 . . an )dn|r /n . For r = 2, when any square matrix can be diagonalized, determinant is always reduced to the one of diagonal matrices.

5 Koszul complex. II. Explicit expression for determinant of exact complex Now, after examples are considered, we can return to generic case. The case of a two-term complex: We have two rectangular matrices, p × q and q × r, Aia and Bαa . ip Aia˜11 . . αr Bαa11 . . 40) Here the last epsilon has exactly p + r = q indices and R is our resultant. s. in power sign(p − r) = ±1. If p = r, then R = 1. 40) it follows that the resultant can be considered as irreducible factor of any maximal-size minor of matrix A (if p > r) of B (if p < r).

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