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For example, to solve a system of n equations for n unknowns by performing row operations on the matrix until it is in echelon form, and then solving for each unknown in reverse order, requires n(n + 1)/2 divisions, (2n 3 + 3n 2 − 5n)/6 multiplications, and (2n 3 + 3n 2 − 5n)/6 subtractions, [10] for a total of approximately 2n 3 /3 operations.
A system of linear equations is said to be in row echelon form if its augmented matrix is in row echelon form. Similarly, a system of linear equations is said to be in reduced row echelon form or in canonical form if its augmented matrix is in reduced row echelon form. The canonical form may be viewed as an explicit solution of the linear system.
These results may be easier to understand by putting the augmented matrix of the coefficients of the system in row echelon form by using Gaussian elimination. This row echelon form is the augmented matrix of a system of equations that is equivalent to the given system (it has exactly the same solutions).
The cost of solving a system of linear equations is approximately floating-point operations if the matrix has size . This makes it twice as fast as algorithms based on QR decomposition , which costs about 4 3 n 3 {\textstyle {\frac {4}{3}}n^{3}} floating-point operations when Householder reflections are used.
Comments: The LUP and LU decompositions are useful in solving an n-by-n system of linear equations =. These decompositions summarize the process of Gaussian elimination in matrix form. Matrix P represents any row interchanges carried out in the process of Gaussian elimination.
In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as
The use of Gaussian elimination for putting the augmented matrix in reduced row echelon form does not change the set of solutions and the ranks of the involved matrices. The theorem can be read almost directly on the reduced row echelon form as follows. The rank of a matrice is number of nonzero rows in its reduced row echelon form.
Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform.