LinearSolve
LinearSolve[m,b]
finds an x that solves the matrix equation m.x==b.
LinearSolve[m]
generates a LinearSolveFunction[…] that can be applied repeatedly to different b.
Details and Options
 LinearSolve works on both numerical and symbolic matrices, as well as SparseArray objects.
 The argument b can be either a vector or a matrix. »
 The matrix m can be square or rectangular. »
 LinearSolve[m] and LinearSolveFunction[…] provide an efficient way to solve the same approximate numerical linear system many times.
 LinearSolve[m,b] is equivalent to LinearSolve[m][b].
 For underdetermined systems, LinearSolve will return one of the possible solutions; Solve will return a general solution. »
 LinearSolve has the following options and settings:

Method Automatic method to use Modulus 0 whether to solve modulo n ZeroTest Automatic test to determine when expressions are zero  The ZeroTest option only applies to exact and symbolic matrices.
 With Method>Automatic, the method is automatically selected depending upon input.
 Explicit Method settings for exact and symbolic matrices include:

"CofactorExpansion" Laplace cofactor expansion "DivisionFreeRowReduction" Bareiss method of divisionfree row reduction "OneStepRowReduction" standard row reduction  Explicit Method settings for approximate numeric matrices include:

"Banded" banded matrix solver "Cholesky" Cholesky method for positive definite Hermitian matrices "Krylov" iterative Krylov sparse solver "Multifrontal" direct sparse LU decomposition "Pardiso" parallel direct sparse solver
Examples
open allclose allBasic Examples (3)
Scope (13)
Basic Uses (7)
Solve a case where is a matrix:
Find a solution for an exact, rectangular matrix:
Compute a solution at arbitrary precision:
Solve the system when is a matrix:
Solve for when is a matrix of different dimensions:
When no right‐hand side for is given, a LinearSolveFunction is returned:
This contains data to solve the problem quickly for a few values of :
Special Matrices (6)
As the result is typically not sparse, the result is returned as an ordinary list:
Sparse methods are used to efficiently solve sparse matrices:
Solve a system with structured matrices:
Use a different type of matrix structure:
An identity matrix always produces a trivial solution:
Solve a linear system whose coefficient matrix is a Hilbert matrix:
Solve a system whose coefficients are univariate polynomials of degree :
Options (7)
Method (6)
"Cholesky" (1)
"Krylov" (2)
The following suboptions can be specified for the method "Krylov":
"BasisSize"  the size of the Krylov basis (GMRES only)  
"MaxIterations"  the maximum number of iterations  
"Method"  methods to be used  
"Preconditioner"  which preconditioner to apply  
"PreconditionerSide"  how to apply a preconditioner ("Left" or "Right")  
"ResidualNormFunction"  A norm function that computes a norm of the residual of the solution  
"StartingVector"  the initial vector to start iterations  
"Tolerance"  the tolerance used to terminate iterations 
Possible settings for "Method" include:
"BiCGSTAB"  iterative method for arbitrary square matrices  
"ConjugateGradient"  iterative method for Hermitian positive definite matrices  
"GMRES"  iterative method for arbitrary square matrices 
Possible settings for "Preconditioner" include:
"ILU0"  a preconditioner based on an incomplete LU factorization of the original matrix without fillin  
"ILUT"  a variant of ILU0 with fillin  
"ILUTP"  a variant of ILUT with column permutation 
Possible suboptions for "Preconditioner" include:
"FillIn"  upper bound on the number of additional nonzero elements in a row introduced by the ILUT preconditioner  
"PermutationTolerance"  when to permute columns  
"Tolerance"  drop tolerance (any element of magnitude smaller than this tolerance is treated as zero) 
"Multifrontal" (1)
Applications (11)
Spans and Linear Independence (3)
The following three vectors are not linearly independent:
The equation with a generic righthand side does not have a solution:
Equivalently, the equation with the identity matrix on the righthand side has no solution:
The following three vectors are linearly independent:
The equation with a generic righthand side has a solution:
Equivalently, the equation with the identity matrix on the righthand side has a solution:
The solution is the inverse of :
Determine if the following vectors are linearly independent or not:
As does not have a solution for an arbitrary , they are not linearly independent:
Equation Solving and Invertibility (6)
Solve the following system of equations:
Rewrite the system in matrix form:
Use LinearSolve to find a solution:
Show that the solution is unique using NullSpace:
Verify the result using SolveValues:
Find all solutions of the following system of equations:
First, write the coefficient matrix , variable vector and constant vector :
LinearSolve gives a particular solution:
NullSpace gives a basis for solutions to the homogeneous equation :
Define to be an arbitrary linear combination of the elements of :
The general solution is the sum of and :
Determine if the following matrix has an inverse:
Since the system has no solution, does not have an inverse:
Verify the result using Inverse:
Determine if the following matrix has a nonzero determinant:
Since the system has a solution, 's determinant must be nonzero:
Confirm the result using Det:
Find the inverse of the following matrix:
To find the inverse, first solve the system :
Verify the result using Inverse:
Solve the system , with several different by means of computing a LinearSolveFunction:
Perform the computation by inverting the matrix and multiplying by the inverse:
The results are practically identical, even though LinearSolveFunction is multiple times faster:
Calculus (2)
Newton's method for finding a root of a multivariate function:
Compare with the answer found by FindRoot:
Approximately solve the boundary value problem using discrete differences:
Properties & Relations (9)
For an invertible matrix , LinearSolve[m,b] gives the same result as SolveValues for the corresponding system of equations:
Create the corresponding system of linear equations:
Confirm that SolveValues gives the same result:
LinearSolve always returns the trivial solution to the homogenous equation :
Use NullSpace to get the complete spanning set of solutions if is singular:
Compare with the result of SolveValues:
If is nonsingular, the solution of is the inverse of when is the identity matrix:
In this case there is no solution to :
Use LeastSquares to minimize :
Compare to general minimization:
If can be solved, LeastSquares is equivalent to LinearSolve:
For a square matrix, LinearSolve[m,b] has a solution for a generic b iff Det[m]!=0:
For a square matrix, LinearSolve[m,b] has a solution for a generic b iff m has full rank:
For a square matrix, LinearSolve[m,b] has a solution for a generic b iff m has an inverse:
For a square matrix, LinearSolve[m,b] has a solution for a generic b iff m has a trivial null space:
Possible Issues (3)
Solution found for an underdetermined system is not unique:
All solutions are found by Solve:
LinearSolve gave the solution corresponding to :
With illconditioned matrices, numerical solutions may not be sufficiently accurate:
The solution is more accurate if sufficiently high precision is used:
Some of the linear solvers available are not deterministic. Set up a system of equations:
The "Pardiso" solver is not deterministic:
The Automatic solver method is deterministic:
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