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In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations ˙ = () is a matrix-valued function () whose columns are linearly independent solutions of the system. [1]
The eight-point algorithm is an algorithm used in computer vision to estimate the essential matrix or the fundamental matrix related to a stereo camera pair from a set of corresponding image points. It was introduced by Christopher Longuet-Higgins in 1981 for the case of the essential matrix.
on an interval I of the real line, where A(t) for t ∈ I denotes a square matrix of dimension n with real or complex entries. Let Φ denote a matrix-valued solution on I, meaning that Φ(t) is the so-called fundamental matrix, a square matrix of dimension n with real or complex entries and the derivative satisfies
The ideas behind the MFS were developed primarily by V. D. Kupradze and M. A. Alexidze in the late 1950s and early 1960s. [1] However, the method was first proposed as a computational technique much later by R. Mathon and R. L. Johnston in the late 1970s, [2] followed by a number of papers by Mathon, Johnston and Graeme Fairweather with applications.
In mathematics, the Wronskian of n differentiable functions is the determinant formed with the functions and their derivatives up to order n – 1.It was introduced in 1812 by the Polish mathematician Józef Wroński, and is used in the study of differential equations, where it can sometimes show the linear independence of a set of solutions.
Once the fundamental solution is found, it is straightforward to find a solution of the original equation, through convolution of the fundamental solution and the desired right hand side. Fundamental solutions also play an important role in the numerical solution of partial differential equations by the boundary element method .
So there is a unique solution to the original system of equations. Instead of stopping once the matrix is in echelon form, one could continue until the matrix is in reduced row echelon form, as it is done in the table. The process of row reducing until the matrix is reduced is sometimes referred to as Gauss–Jordan elimination, to distinguish ...
This fundamental matrix will provide the homogeneous solution, and if added to a particular solution will give the general solution to the inhomogeneous equation. Let x = X y {\displaystyle x=Xy\,} be the general solution.