What are Kyle Numbers and how do you compute them?

[mahrap]

My linear algebra uses "Kyle Numbers" to compute some kernels. But it does not elaborate on what they are and how they are used to compute the kernel? Please help.

 

[fzero]

I found http://www.math.harvard.edu/archive/21b_spring_09/faq.html. The name isn't standard, but the concept seems to make sense. Post back if the explanation on that page is insufficient.

 

[mahrap]

I found http://www.math.harvard.edu/archive/21b_spring_09/faq.html. The name isn't standard, but the concept seems to make sense. Post back if the explanation on that page is insufficient.

I understand the concept. But do you think there is an algorithm to compute these numbers? Or should I just find the kernel by sticking with row reduction?

 

[fzero]

I understand the concept. But do you think there is an algorithm to compute these numbers? Or should I just find the kernel by sticking with row reduction?

It would be fine to stick with row reduction. For 2x2 matrices with rational entries, you could come up with an algorithm to find the smallest integers $(k_1,k_2)$ such that the column vectors satisfy $k_1 u_1 + k_2 u_2 =0.$ Those would be the Kyle numbers. For a larger matrix, you can have similar relationships, but it gets more and more complicated. I would end up using row reduction to find the coefficients of the linear combination myself.

 

[blue_raver22]

Kyle Numbers, also known as Kyle's Numbers, are a set of numbers used in linear algebra to compute certain kernels. These numbers were introduced by mathematician Kyle in his research on matrix theory. They are a set of real numbers that are used as coefficients in the computation of kernels.

To understand how Kyle Numbers are used to compute kernels, we first need to understand what a kernel is. In linear algebra, a kernel is a vector space that represents the set of all vectors that are mapped to the zero vector by a linear transformation. In simple terms, it is the set of all solutions to a system of linear equations that have a zero output.

Now, to compute a kernel, we need to find the solutions to the system of linear equations. This is where Kyle Numbers come into play. These numbers are used as coefficients in the system of equations, and by manipulating them, we can find the solutions to the system.

For example, let's say we have a system of linear equations represented as Ax = 0, where A is a matrix and x is a vector. To compute the kernel, we need to find the values of x that satisfy this equation. By using Kyle Numbers as coefficients in the matrix A, we can manipulate the equations to find the values of x.

In summary, Kyle Numbers are a set of real numbers used as coefficients in the computation of kernels in linear algebra. They are an essential tool in solving systems of linear equations and have various applications in mathematics and other fields of science.