Solving Singular matrices

In summary: These algorithms can be implemented in a code, and are commonly used in computations involving matrices.
  • #1
nelectrode
10
0
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
 
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  • #2
nelectrode said:
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.
nelectrode said:
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.

nelectrode said:
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.
 
  • #3
thanks a lot :smile:
 
  • #4
nelectrode said:
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.
 

1. What is a singular matrix?

A singular matrix is a square matrix that does not have an inverse. This means that it cannot be inverted or solved for a unique solution.

2. Why is it important to solve singular matrices?

Solving singular matrices is important because it allows us to find solutions to systems of linear equations. These solutions can provide important insights and predictions in various fields such as engineering, economics, and physics.

3. How can I identify a singular matrix?

A singular matrix can be identified by calculating its determinant. If the determinant is equal to 0, then the matrix is singular. Another way to identify a singular matrix is by looking at its rank. A matrix with rank less than its dimensions is singular.

4. What methods can be used to solve singular matrices?

There are several methods that can be used to solve singular matrices, such as Gaussian elimination, LU decomposition, and singular value decomposition. These methods involve manipulating the matrix through row operations to reduce it to a form that can be solved.

5. Can a singular matrix be used to solve real-world problems?

Yes, singular matrices can be used to solve real-world problems. In fact, they are commonly used in various fields such as economics, physics, and engineering to model and predict complex systems. However, it is important to note that the solutions obtained from solving a singular matrix may not always be accurate or unique, so caution must be taken when interpreting the results.

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