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In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
In linear algebra, a Jordan normal form, also known as a Jordan canonical form, [1] [2] is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis.
Given a module M over a ring R, a linear form on M is a linear map from M to R, where the latter is considered as a module over itself. The space of linear forms is always denoted Hom k (V, k), whether k is a field or not. It is a right module if V is a left module.
This glossary of linear algebra is a list of definitions and terms relevant to the field of linear algebra, the branch of mathematics concerned with linear equations and their representations as vector spaces. For a glossary related to the generalization of vector spaces through modules, see glossary of module theory
On the other hand, this makes the Frobenius normal form rather different from other normal forms that do depend on factoring the characteristic polynomial, notably the diagonal form (if A is diagonalizable) or more generally the Jordan normal form (if the characteristic polynomial splits into linear factors). For instance, the Frobenius normal ...
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any m × n {\displaystyle m\times n} matrix.
In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z.Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in R n, the Hermite normal form can solve problems about the solution to the linear system Ax=b where this time x is restricted to have integer coordinates only.
He developed MATLAB's initial linear algebra programming in 1967 with his one-time thesis advisor, George Forsythe. [21] This was followed by Fortran code for linear equations in 1971. [21] Before version 1.0, MATLAB "was not a programming language; it was a simple interactive matrix calculator. There were no programs, no toolboxes, no graphics.