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Linear Algebra for Quantum Mechanics
Published in Caio Lima Firme, Quantum Mechanics, 2022
The Laplace expansion is a method to obtain the determinant of a matrix based on cofactors minor matrix. The cofactors, Cij, of the square matrix A are (−1)i+j times the determinant of the submatrix Aij, D(Aij), obtained from A by deleting ith rows and jth columns of A. The D(Aij) is called minor Mij of element aij of a determinant D obtained by deleting row i and column j The cofactors of A form a new matrix called cofactor matrix, C, whose elements are: Cij=(−1)i+j·D(Aij)Cij=(−1)i+j·Mij
Introduction to Linear Algebra
Published in Timothy Bower, ®, 2023
The classic technique for computing a determinant is called the Laplace expansion. The complexity of computing a determinant by the classic Laplace expansion method has run-time complexity on the order of n! (n–factorial), which is fine for small matrices, but is too slow for large (n>3) matrices.
Solution of the ‘sign problem’ in the path integral Monte Carlo simulations of strongly correlated Fermi systems: thermodynamic properties of helium-3
Published in Molecular Physics, 2022
V. S. Filinov, R. S. Syrovatka, P. R. Levashov
For better understanding of the mathematical structure of Equation (4), let us consider the density matrix of two spinless fermions (): At weak degeneracy, the main contribution to the partition function comes from the pair exchange interactions (being the pair Fermi repulsion, see Equation (5) for ) [25]. As a consequence the Gram determinant in Equation (4) can be approximated by the product of two-particle determinants () (according to the Laplace expansion for N-particle determinant), providing the positive semidefiniteness of the obtained approximation (4).
trlib : a vector-free implementation of the GLTR method for iterative solution of the trust region problem
Published in Optimization Methods and Software, 2018
F. Lenders, C. Kirches, A. Potschka
For successive pL iterations, the fact that the tridiagonal matrices grow by one column and row in each iteration may be exploited to save most of the computational effort involved. As noted by Parlett and Reid [42], the recurrence to compute the via Cholesky decomposition of in Definition 9 is identical with the recurrence that results from applying a Laplace expansion for the determinant of tridiagonal matrices [16, § 2.1.4]. Comparing the recurrences thus yields the explicit formula where denotes the principal submatrix of T obtained by erasing the last column and row, and and enumerate the eigenvalues of T and , respectively. The right-hand side is obtained by identifying numerator and denominator with the characteristic polynomials of T and , and by factorizing these.
Range sets for weak efficiency in multiobjective linear programming and a parametric polytopes intersection problem
Published in Optimization, 2018
Milan Hladík, Miroslav Rada, Sebastian Sitarz, Elif Garajová
Now, assume s=n. Without loss of generality assume that , which can be achieved by a shift. The normal vector of the plane containing is computed by the Laplace expansion of the following determinant along the last row The absolute term is . Thus, equation , which draws , is a polynomial of t of degree s and therefore has at most s disconnected roots.