How to Solve Equations with a Singular Matrix?

[nelectrode]

Hi,

How would you solve a singular matrix? ie when determinant is zero.

Lets assume an equation, (1) Ax+By=E and (2) Cx+Dy=F

if the determinant; AD-BC = 0, and therefore the matrix is singular, How to go around solving the equation?

LU decomposition, Gaussian elimination?

Ideally I am looking for a method which could be "easily" implemented in a code.

Thanks

 

[Mark44]

Hi,

How would you solve a singular matrix? ie when determinant is zero.

Lets assume an equation, (1) Ax+By=E and (2) Cx+Dy=F

That is a system of two equations (not one) in two unknowns, so presumably your matrix is 2 x 2.

if the determinant; AD-BC = 0, and therefore the matrix is singular, How to go around solving the equation?

The system might have no solution or it might have an infinite number of solutions.
A couple of examples might help to shed some light here.
Example 1.
x + 2y = 3
2x + 4y = 6
The equations in this system are equivalent, so geometrically the two equations represent a single line. Here there are an infinite number of solutions. Each point on the first line is also on the second line.

Example 2.
x + 2y = 3
2x + 4y = 1
The equations in this system represent two parallel lines with no common point of intersection. The system has no solutions.

LU decomposition, Gaussian elimination?

Ideally I am looking for a method which could be "easily" implemented in a code.

Assuming that your systems consist of two equations in two unknowns, I would focus my efforts on those systems for which the discriminant is nonzero (i.e., the systems that have a unique solution). Once you determine that the discriminant is nonzero, you could use Cramer's Rule to determine the solution.

If the discriminant is zero, I don't see any point in trying to use Gaussian elimination or LU decomposition. In a system of two equations with two unknowns for which the discrimant is zero, there will either be an infinite number of solutions or no solution at all.

 

[nelectrode]

thanks a lot

 

[Stephen Tashi]

How to go around solving the equation?

LU decomposition, Gaussian elimination?

Ideally I am looking for a method which could be "easily" implemented in a code.

Look up algorithms for computing the "generalized inverse" of a matrix.