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Solving Linear Systems
Many calculations
involve solving systems of linear equations.
the equations explicitly,
In many cases, you will find it convenient
to write down
and then solve them using Solve.
In some cases, however,
you may prefer to convert the system of linear equations into a matrix equation,
apply matrix manipulation
operations
part of a general algorithm,
and then
to solve it. This approach is often useful when the system of equations arises as
and you do not know in advance how many variables will be involved.
A system of linear equations can be stated in matrix form as m.x = b, where x is the vector of variables.
Note that if your system of equations is sparse, so that most of the entries in the matrix
m are zero, then it is best to
represent the matrix as a SparseArray object. As discussed in "Sparse Arrays: Linear Algebra" , you can convert from
symbolic
equations
to SparseArray objects
using
CoefficientArrays. All the functions
described
here work on
SparseArray objects as well as ordinary matrices.
LinearSolve@m,bD
a vector x which solves the matrix equation m.x == b
NullSpace@mD
a list of linearly independent vectors whose linear combinations
solutions to the matrix equation m.x == 0
MatrixRank@mD
the number of linearly independent
RowReduce@mD
a simplified form of m obtained by making linear combinations
span all
rows or columns of m
of rows
Solving and analyzing linear systems.
Here is a 2x2 matrix.
In[1]:=
Out[1]=
m = 881, 5<, 82, 1<<
881, 5<, 82, 1<<
This gives two linear equations.
In[2]:=
Out[2]=
m.8x, y< == 8a, b<
8x + 5 y, 2 x + y< Š 8a, b<
You can use Solve directly to solve these equations.
In[3]:=
Solve@%, 8x, y<D
Out[3]=
::x ®
1
H-a + 5 bL, y ®
9
1
H2 a - bL>>
9
You can also get the vector of solutions by calling LinearSolve. The result is equivalent to the one you get from Solve.
In[4]:=
LinearSolve@m, 8a, b<D
Out[4]=
:
1
9
H-a + 5 bL,
1
H2 a - bL>
9
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Another way to solve the equations is to invert the matrix m, and then multiply 8a, b< by the inverse. This is not as efficient as using
LinearSolve.
In[5]:=
[email protected], b<
Out[5]=
:-
a
5b
2a
,
+
9
9
b
-
9
>
9
RowReduce performs a version of Gaussian elimination and can also be used to solve the equations.
In[6]:=
RowReduce@881, 5, a<, 82, 1, b<<D
Out[6]=
::1, 0,
1
H-a + 5 bL>, :0, 1,
9
1
H2 a - bL>>
9
If you have a square matrix
m with a non -zero determinant,
equation m.x = b for any b. If, however, the matrix
then you can always find a unique solution to the matrix
m has determinant
infinite number of vectors x which satisfy m.x = b for a particular
zero, then there may be either no vector, or an
b. This occurs when the linear equations embodied in
m are not independent.
When m has determinant
of vectors
x satisfying
zero, it is nevertheless
this equation
expressed as a linear combination
always possible to find non -zero vectors x that satisfy m.x = 0. The set
form the null space or kernel of the matrix
of a particular
m. Any of these vectors
can be
set of basis vectors, which can be obtained using NullSpace@mD.
Here is a simple matrix, corresponding to two identical linear equations.
In[7]:=
Out[7]=
m = 881, 2<, 81, 2<<
881, 2<, 81, 2<<
The matrix has determinant zero.
In[8]:=
Det@mD
Out[8]=
0
LinearSolve cannot find a solution to the equation m.x Š b in this case.
In[9]:=
LinearSolve@m, 8a, b<D
Out[9]=
LinearSolve@881, 2<, 81, 2<<, 8a, b<D
LinearSolve::nosol : Linear equation encountered that has no solution. ‡
There is a single basis vector for the null space of m.
In[10]:=
Out[10]=
NullSpace@mD
88-2, 1<<
Multiplying the basis vector for the null space by m gives the zero vector.
In[11]:=
Out[11]=
m.%@@1DD
80, 0<
There is only 1 linearly independent row in m.
In[12]:=
MatrixRank@mD
Out[12]=
1
NullSpace and MatrixRank have to determine
whether
particular
combinations
of matrix
elements
are zero. For
numerical
Tolerance
option can be
©approximate
1988– 2007 Wolfram
Research, matrices,
Inc. All rights the
reserved.
http://reference.wolfram.com
used to specify how close to zero is considered good
enough.
sometimes
For
exact
symbolic
matrices,
you
may
need
to
specify
something
ZeroTest -> HFullSimplify@ðD == 0 &L to force more to be done to test whether symbolic expressions are zero.
like
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NullSpace and MatrixRank have to determine
approximate
enough.
numerical
For
matrices,
exact
whether
particular
combinations
of matrix
- Complete Documentation
elements
are zero. For
the Tolerance option can be used to specify how close to zero is considered good
symbolic
matrices,
you
may
sometimes
need
to
specify
something
ZeroTest -> HFullSimplify@ðD == 0 &L to force more to be done to test whether symbolic expressions are zero.
like
Here is a simple symbolic matrix with determinant zero.
In[13]:=
Out[13]=
m = 88a, b, c<, 82 a, 2 b, 2 c<, 83 a, 3 b, 3 c<<
88a, b, c<, 82 a, 2 b, 2 c<, 83 a, 3 b, 3 c<<
The basis for the null space of m contains two vectors.
In[14]:=
NullSpace@mD
Out[14]=
::-
c
, 0, 1>, :-
a
b
, 1, 0>>
a
Multiplying m by any linear combination of these vectors gives zero.
In[15]:=
Out[15]=
[email protected] %@@1DD + y %@@2DDLD
80, 0, 0<
An important
feature
of functions
like LinearSolve and NullSpace is that they work with rect angular , as well as
square, matrices.
When you represent a system of linear equations by a matrix equation of the form m.x = b, the number of columns in m
gives the number of variables, and the number of rows gives the number of equations. There are a number of cases.
Unde r de t e r m ine d
number of equations less than the number of variables;
many solutions may exist
Ove r de t e r m ine d
number of equations more than the number of variables;
or may not exist
N onsingula r
number of independent equations equal to the number of variables, and
determinant non-zero; a unique solution exists
Consist e nt
I nconsist e nt
at least one solution exists
no solutions or
solutions may
no solutions exist
Classes of linear systems represented by rectangular matrices.
This asks for the solution to the inconsistent set of equations x = 1 and x = 0.
In[16]:=
LinearSolve@881<, 81<<, 81, 0<D
Out[16]=
LinearSolve@881<, 81<<, 81, 0<D
LinearSolve::nosol : Linear equation encountered that has no solution. ‡
This matrix represents two equations, for three variables.
In[17]:=
Out[17]=
m = 881, 3, 4<, 82, 1, 3<<
881, 3, 4<, 82, 1, 3<<
LinearSolve gives one of the possible solutions to this underdetermined set of equations.
In[18]:=
Out[18]=
v = LinearSolve@m, 81, 1<D
2 1
: , , 0>
5 5
When a matrix represents an underdetermined system of equations, the matrix has a non-trivial null space. In this case, the null space is
spannedResearch,
by a single
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Inc.vector.
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When a matrix represents an underdetermined system of equations, the matrix has a non-trivial null space. In this case, the null space is
spanned by a single vector.
In[19]:=
Out[19]=
NullSpace@mD
88-1, -1, 1<<
If you take the solution you get from LinearSolve, and add any linear combination of the basis vectors for the null space, you still get a
solution.
In[20]:=
Out[20]=
m.Hv + 4 %@@1DDL
81, 1<
The number of independent
equations is the rank of the matrix MatrixRank@mD. The number of redundant
Length@NullSpace@mDD. Note that the sum of these quantities
equations is
is always equal to the number of columns in m.
generate a function for solving equations of the form m.x = b
LinearSolve@mD
Generating LinearSolveFunction objects.
In some applications,
you will want to solve equations of the form m.x = b many times with the same m, but different
You can do this efficiently
in Mat hem at ica by using LinearSolve@mD to create a single LinearSolveFunction that you
can apply to as many vectors as you want.
This creates a LinearSolveFunction.
In[21]:=
f = LinearSolve@881, 4<, 82, 3<<D
Out[21]=
LinearSolveFunction@82, 2<, <>D
You can apply this to a vector.
In[22]:=
f@85, 7<D
Out[22]=
:
13
3
,
5
>
5
You get the same result by giving the vector as an explicit second argument to LinearSolve.
In[23]:=
LinearSolve@881, 4<, 82, 3<<, 85, 7<D
Out[23]=
:
13
3
,
5
>
5
But you can apply f to any vector you want.
In[24]:=
f@8-5, 9<D
Out[24]=
:
51
19
,-
5
>
5
LeastSquares@m,bD
b.
give a vector x that solves the least -squares problem m.x == b
Solving least-squares problems.
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This linear system is inconsistent.
In[25]:=
LinearSolve@881, 2<, 83, 4<, 85, 6<<, 8-1, 0, 2<D
LinearSolve::nosol : Linear equation encountered that has no solution. ‡
Out[25]=
LinearSolve@881, 2<, 83, 4<, 85, 6<<, 8-1, 0, 2<D
LeastSquares finds a vector x that minimizes m.x - b in the least-squares sense.
In[26]:=
Out[26]=
LeastSquares@881, 2<, 83, 4<, 85, 6<<, 8-1, 0, 2<D
8
23
: ,>
3
12
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- Complete Documentation
5