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The Gram matrix is symmetric in the case the inner product is real-valued; it is Hermitian in the general, complex case by definition of an inner product. The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can ...
The Gram matrix of a sequence of points ,, …, in k-dimensional space ℝ k is the n×n matrix = of their dot products (here a point is thought of as a vector from 0 to that point): g i j = x i ⋅ x j = ‖ x i ‖ ‖ x j ‖ cos θ {\displaystyle g_{ij}=x_{i}\cdot x_{j}=\|x_{i}\|\|x_{j}\|\cos \theta } , where θ {\displaystyle \theta ...
with N B and N G being the molarities 1-column matrices of the batch and glass components respectively, and B being the batching matrix. [1] [2] [3] The symbol "T" stands for the matrix transpose operation, "−1" indicates matrix inversion, and the sign "·" means the scalar product.
The first two steps of the Gram–Schmidt process. In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other.
After the algorithm has converged, the singular value decomposition = is recovered as follows: the matrix is the accumulation of Jacobi rotation matrices, the matrix is given by normalising the columns of the transformed matrix , and the singular values are given as the norms of the columns of the transformed matrix .
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
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The mean-squared density matrices, representing deviation from the true density matrix, may be rewritten as a Gram matrix, and an iterative algorithm is used to minimize the Gram matrix and solve the QP. [17] [6]