This is a little howto for using matrices to solve simultaneous linear equations. Don't let it be confusing, you can just plug your values into the examples to get answers. I have included a bit of theory, so that you are not "confused" by unexpected results.

When you solve simultaneous equations, it is easiest to use matrix equations.

Generally the solution goes like this: Say you have a matrix equation

Ax=b

where A is a 'n x n' matrix, x is a 'n x 1' matrix (vector), and b is a set of known values of 'n x 1' then the solution is to find the inverse of the matrix A and mutiply it by b

x = inv(A) * b

This all corresponds with solving simultaneous linear equations. You need n number of equations to solve for n variables.

There is only one glitch.... there is no inverse to a matrix, if there is not a unique solution. And there is no inverse, if the determinant=0. So checking the determinant is an easy way of telling whether you will get a unique solution. Just because you don't get a unique solution, dosn't mean there are not alot of good solutions. In Math::Matrix::Real, you can use the $dim variable to see what form your solutions is. If $dim=0, the solution is a unique point. If $dim=1, the solution is a line. If $dim=2, the solution is a plane. etc...up into mind-boggling spaces.

The matrix inverse method is "mathematically pure", but not always easy to implement in real solutions. Most matrix solutions use a method called LR-decomposition. It is well explained in perldoc Math::Matrix::Real.

So here are 2 examples, where I solve the same set of equations, with Math::Matrix, and Math::MatrixReal. The first is not unique, where the answer is a plane, and you can see where some confusion can be generated, when Math::Matrix gives different answers from Math::MatrixReal.So always check the determinant or the $dim variable, so see if the solution is unique. (I've mixed my matrix loading notation to show you the equivalence of the different available methods).

EXAMPLE 1

####################################################################

With Math::Matrix 

#!/usr/bin/perl
use warnings;
use Math::Matrix;

#say you have equations
# 1x   2y   3z   4w  = 17
# 5x   6y   7z   8w  = 18
# 9x   10y  11z  12w = 19
# 13x  14y  15z  16w = 20

$a = new Math::Matrix ([1,2,3,4],
                       [5,6,7,8],
                       [9,10,11,12],
                       [13,14,15,16]);

$x = new Math::Matrix ([17,18,19,20]);

print "det-> ",$a->determinant,"\n";

$a->print("A\n");
$E = $a->concat($x->transpose);
$E->print("Equation system\n");
$s = $E->solve;
$s->print("Solutions s\n");
$a->multiply($s)->print("A*s\n");

[download]
OUTPUT
det-> 0
A
1.00000    2.00000    3.00000    4.00000
5.00000    6.00000    7.00000    8.00000
9.00000   10.00000   11.00000   12.00000
13.00000  14.00000   15.00000   16.00000

Equation system
1.00000    2.00000    3.00000    4.00000   17.00000
5.00000    6.00000    7.00000    8.00000   18.00000
9.00000   10.00000   11.00000   12.00000   19.00000
13.00000  14.00000   15.00000   16.00000   20.00000

Solutions s
-16.50000
16.75000
0.00000
0.00000

A*s
17.00000
18.00000
19.00000
20.00000

[download]
###########################################################

with Math::MatrixReal 

#!/usr/bin/perl
use warnings;
use strict;
use Math::MatrixReal;

my ($a1, $b1, $c1, $d1, $e1,
$a2, $b2, $c2, $d2, $e2,
$a3, $b3, $c3, $d3, $e3,
$a4, $b4, $c4, $d4, $e4) =
(1,  2,  3,  4,  17,
 5,  6,  7,  8,  18,
 9,  10, 11, 12, 19,
 13, 14, 15, 16, 20);

my $A = Math::MatrixReal->new_from_string(<<EOF);
[ $a1 $b1 $c1 $d1 ]
[ $a2 $b2 $c2 $d2 ]
[ $a3 $b3 $c3 $d3 ]
[ $a4 $b4 $c4 $d4 ]
EOF

my $b = Math::MatrixReal->new_from_string(<<EOF);
[ $e1 ]
[ $e2 ]
[ $e3 ]
[ $e4 ]
EOF

print "det-> ",$A->det(),"\n";

my $LR = $A->decompose_LR();

my ($dim, $solution, $B) = $LR->solve_LR($b);

#print "dim->$dim\n\nsolution->\n$solution\n\nbase->\n$B\n\n";
print "dim->$dim\n\nsolution->\n$solution\n\nbase->\n$B\n\n";

my $out = $A * $solution;
print $out,"\n";

[download]
OUTPUT:
det-> 0
dim->2

solution->
[ -5.333333333333E+00 ]
[  0.000000000000E+00 ]
[  0.000000000000E+00 ]
[  5.583333333333E+00 ]


[  1.700000000000E+01 ]
[  1.800000000000E+01 ]
[  1.900000000000E+01 ]
[  2.000000000000E+01 ]

[download]
############################################################

So you can see some confusion can be generated here. 2 different answers are generated, but they both are good, since the solutions are not unique, as evidenced by $dim = 2, and determinant=0. This means that the answer is an entire set of points which make a plane.

##############################################################

So here is another example(similar) except the solution is unique.

EXAMPLE 2

########################################################

With Math::Matrix 

#!/usr/bin/perl
use warnings;
use Math::Matrix;

$a = new Math::Matrix ([1, -2,  3,  4],
                       [4, -5,  6,  7],
                       [7,  8, -9,  10],
                       [8,  9, -9, -11],
                       );

$x = new Math::Matrix ([17,18,19,20]);

print "det-> ",$a->determinant,"\n";

$a->print("A\n");
$E = $a->concat($x->transpose);
$E->print("Equation system\n");
$s = $E->solve;
$s->print("Solutions s\n");
$a->multiply($s)->print("A*s\n");

[download]
OUTPUT:
det-> -1104

A
1.00000   -2.00000    3.00000    4.00000
4.00000   -5.00000    6.00000    7.00000
7.00000    8.00000   -9.00000   10.00000
8.00000    9.00000   -9.00000  -11.00000

Equation system
1.00000   -2.00000    3.00000    4.00000   17.00000
4.00000   -5.00000    6.00000    7.00000   18.00000
7.00000    8.00000   -9.00000   10.00000   19.00000
8.00000    9.00000   -9.00000  -11.00000   20.00000

Solutions s
1.45652
18.03261
16.02899
0.88043

A*s
17.00000
18.00000
19.00000
20.00000

[download]
#######################################################

With Math::MatrixReal: 

#!/usr/bin/perl
use warnings;
use strict;
use Math::MatrixReal;

my ($a1, $b1, $c1, $d1, $e1,
$a2, $b2, $c2, $d2, $e2,
$a3, $b3, $c3, $d3, $e3,
$a4, $b4, $c4, $d4, $e4) =
(1,  -2,   3,   4,  17,
 4,  -5,   6,   7,  18,
 7,   8,  -9,  10,  19,
 8,   9,  -9, -11,  20);


my $A = Math::MatrixReal->new_from_string(<<EOF);
[ $a1 $b1 $c1 $d1 ]
[ $a2 $b2 $c2 $d2 ]
[ $a3 $b3 $c3 $d3 ]
[ $a4 $b4 $c4 $d4 ]
EOF

my $b = Math::MatrixReal->new_from_string(<<EOF);
[ $e1 ]
[ $e2 ]
[ $e3 ]
[ $e4 ]
EOF

print "det-> ",$A->det(),"\n";

my $LR = $A->decompose_LR();

my ($dim, $solution, $B) = $LR->solve_LR($b);

#print "dim->$dim\n\nsolution->\n$solution\n\nbase->\n$B\n\n";
print "dim->$dim\n\nsolution->\n\n$solution\n\n";

my $out = $A * $solution;
print $out,"\n";

[download]
OUTPUT: 
det-> -1104
dim->0

solution->

[  1.456521739130E+00 ]
[  1.803260869565E+01 ]
[  1.602898550725E+01 ]
[  8.804347826087E-01 ]


[  1.700000000000E+01 ]
[  1.800000000000E+01 ]
[  1.900000000000E+01 ]
[  2.000000000000E+01 ]

[download]
########################################################

Ok, now the solution is unique, and Math::Matrix gives the exact same solution as Math::MatrixReal

###########################################################

Finally I just want to demonstrate the validity of the original mathematical method of solving by multiplying both sides of the matrix equation by the matrix inverse. It is basic algebra, in that a matrix times it's inverse equals 1, or in matrix terms, a matrix of all zeros except the diagonal being 1's. Like this: 

[1  0  0  0],
[0  1  0  0],
[0  0  1  0],
[0  0  0  1]

[download]
Also we can use the matrix from the second example, since it's determinant is non-zero, and therefore it has an inverse. 
#!/usr/bin/perl
use warnings;
use Math::MatrixReal;

$a = Math::MatrixReal->new_from_rows ([
[1, -2,  3,   4],
[4, -5,  6,   7],
[7,  8, -9,  10],
[8,  9, -9, -11],
]);

print $a,"\n";

my $b = Math::MatrixReal->new_from_cols ([
[17,18,19,20]
]);
print $b,"\n";

print $a->det(),"\n";

$inva = $a->inverse();
print $inva,"\n";

$c = $inva * $b ;

print "solution->\n$c\n";

[download]
OUTPUT:
[  1.000000000000E+00 -2.000000000000E+00  3.000000000000E+00  4.00000
+0000000E+00 ]
[  4.000000000000E+00 -5.000000000000E+00  6.000000000000E+00  7.00000
+0000000E+00 ]
[  7.000000000000E+00  8.000000000000E+00 -9.000000000000E+00  1.00000
+0000000E+01 ]
[  8.000000000000E+00  9.000000000000E+00 -9.000000000000E+00 -1.10000
+0000000E+01 ]

[  1.700000000000E+01 ]
[  1.800000000000E+01 ]
[  1.900000000000E+01 ]
[  2.000000000000E+01 ]

-1104
[ -1.684782608696E-01  1.739130434783E-01  5.434782608696E-03  5.43478
+2608696E-02 ]
[  1.595108695652E+00 -6.304347826087E-01 -3.532608695652E-02  1.46739
+1304348E-01 ]
[  1.362318840580E+00 -4.492753623188E-01 -8.695652173913E-02  1.30434
+7826087E-01 ]
[  6.793478260870E-02 -2.173913043478E-02  4.619565217391E-02 -3.80434
+7826087E-02 ]

solution->
[  1.456521739130E+00 ]
[  1.803260869565E+01 ]
[  1.602898550725E+01 ]
[  8.804347826087E-01 ]

[download]
#################################################################################


So you can see, that the answers agree, but we can only use the inverse method when the determinant does not equal 0. Corrections, comments, and suggestions welcome. (I'm doing this from memory from my matrix theory 25 years ago :-) )
I'm not really a human, but I play one on earth. flash japh

janitored by ybiC: Balanced <readmore> tags around longish 'examples' section

In reply to Solving Simultaneous Equations with Matrices by zentara

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