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Linear Algebra and its Applications is a biweekly peer-reviewed mathematics journal published by Elsevier and covering matrix theory and finite-dimensional linear algebra. History [ edit ]
Specifically, if A is TU and b is integral, then linear programs of forms like {,} or {} have integral optima, for any c. Hence if A is totally unimodular and b is integral, every extreme point of the feasible region (e.g. { x ∣ A x ≥ b } {\displaystyle \{x\mid Ax\geq b\}} ) is integral and thus the feasible region is an integral polyhedron.
An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map.
Matrices are commonly related to linear algebra. Notable exceptions include incidence matrices and adjacency matrices in graph theory. [1] This article focuses on matrices related to linear algebra, and, unless otherwise specified, all matrices represent linear maps or may be viewed as such.
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
An M-matrix is commonly defined as follows: Definition: Let A be a n × n real Z-matrix.That is, A = (a ij) where a ij ≤ 0 for all i ≠ j, 1 ≤ i,j ≤ n.Then matrix A is also an M-matrix if it can be expressed in the form A = sI − B, where B = (b ij) with b ij ≥ 0, for all 1 ≤ i,j ≤ n, where s is at least as large as the maximum of the moduli of the eigenvalues of B, and I is an ...
The general idea is the following: consider well-known facts of linear algebra and look how to relax the commutativity assumption for matrix elements such that the results will be preserved to be true. The answer is: if and only if M is a Manin matrix. [3] The proofs of all observations is direct 1 line check.
Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra , and play a prominent role in engineering , physics , chemistry , computer science , and economics .
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