Why is a system considered “singular” if there are no solutions / infinite solutions, but “non-singular” if there is a single solution.

Don’t think it was covered in lecture iirc

Why is a system considered “singular” if there are no solutions / infinite solutions, but “non-singular” if there is a single solution.

Don’t think it was covered in lecture iirc

2 Likes

Hello @martis880 and welcome back to the DeepLearning community

A system of equations is said to be “singular” if the coefficient matrix A is not invertible, which means that it has no inverse matrix. If a matrix A is singular, then its determinant is equal to zero.

If a system of linear equations is singular, it means that there is no unique solution to the system. This can happen when the equations are linearly dependent, which means that one equation can be obtained by adding or subtracting multiples of the other equations. In this case, the system is underdetermined, meaning that there are fewer equations than variables, and there may be infinite solutions that satisfy the equations.

If a system of linear equations is non-singular, it means that the coefficient matrix A is invertible and has a unique inverse matrix. This means that there is a unique solution to the system. In this case, the equations are linearly independent, which means that none of the equations can be obtained by adding or subtracting multiples of the other equations. The system is fully determined, meaning that there are as many equations as variables, and there is a unique solution that satisfies all the equations.

Please check out here for more about singular matrices Khan Academy

1 Like

In simple terms, the system of equations is considered singular if the det of the coefficient matrix is equal to 0. it is considered non-singular if det is non- zero. @Isaak_Kamau has further connected it back to what it means in terms of the solution of the system of equations.

@Isaak_Kamau @shanup sorry I wasn’t clear enough in my question.

I understand the difference between singular and non-singular, what I don’t understand is why they were named that way.

Based on the material presented in class, it seems like it would make more sense for them to be named the other way around because a non-singular system of equations (the way it’s defined) has a single solution.

2 Likes

Ok I asked ChatGPT and am satisfied with it’s answer hah, let me know if you guys know of anything else!

I see your point of confusion. The terms “singular” and “nonsingular” can be a bit counterintuitive, especially if you try to interpret them from their common usage in everyday language.

In linear algebra, the term “singular” is not used in its common sense of “single” or “one”. Instead, it refers to the fact that a singular matrix is “special” or “exceptional” in some way, specifically in that it does not have an inverse.

Conversely, a nonsingular matrix is “normal” or “generic” in the sense that it does have an inverse.

In general use, the term “singularity” can have several different meanings depending on the context, but it generally refers to a situation where something is unique, exceptional, or remarkable in some way.

For example, in astronomy and astrophysics, a “singularity” refers to a point in space-time where the laws of physics break down, such as at the center of a black hole.

In technology and computer science, a “technological singularity” refers to a hypothetical future event where artificial intelligence surpasses human intelligence, leading to a rapid and unprecedented acceleration in technological progress.

In mathematics, a “singularity” can refer to a point in a function or a surface where the function or surface is not well-defined, such as a point of discontinuity, a pole, or an essential singularity.

The term “singularity” has its roots in the Latin word “singularis”, which means “unique” or “singular”. The term is used to describe something that is exceptional or different from the norm, often implying a degree of unpredictability or even danger.

In summary, the term “singularity” is used to describe situations that are exceptional, unique, or different from the norm, often implying a degree of unpredictability or even danger. While the term has its roots in the Latin word for “unique”, its modern usage is not necessarily related to the concept of counting or “the count of one”.

5 Likes

Thank you. I had the same query.

thank you for sharing, it was confusing to me too.