What is an example of a well conditioned matrix with a small determinant?

  • Thread starter chadpip
  • Start date
  • Tags
    Matrices
In summary, the conversation involved a question about well-conditioned matrices with small determinants and a previous warning from the forum moderator for posting multiple copies of the same question. The question was ultimately resolved and the thread was left up for further discussion.
  • #1
chadpip
7
0
this is more of a numerical analysis question..so I am not sure where to post it..(also put it up in the computers forum)


im wondering, I've seen examples of ill conditioned matrices with small determinants...but what would be an example of a well conditioned matrix with a very small determinant?
 
Engineering news on Phys.org
  • #2
I've deleted your other posts and issued you a 1-point warning for posting multiple copies of the same question. Do no multiple-post. That is not allowed here on the Physics Forums (PF), or on any other forum system that I'm aware of. I'll leave this one thread here, since it seems like the best match so far for your question.
 
  • #3
oh sorry! i must have overlooked that in the rules :(

but..the good news is i finally figured one out! so forget this question
 

1. What is a well-conditioned matrix?

A well-conditioned matrix is a matrix that has a small condition number, which measures how sensitive the output of a matrix is to small changes in the input. A small condition number indicates that the matrix is stable and the solution to its equations is reliable.

2. How is the condition number of a matrix calculated?

The condition number of a matrix is calculated by taking the ratio of the largest singular value to the smallest singular value. The singular values are calculated using a process called singular value decomposition (SVD).

3. Why is it important for a matrix to be well-conditioned?

A well-conditioned matrix is important because it ensures the stability and accuracy of the solutions to its equations. A poorly conditioned matrix can lead to inaccurate results and errors in calculations.

4. Can a matrix be perfectly well-conditioned?

No, a matrix cannot be perfectly well-conditioned. The condition number of a matrix is always greater than or equal to 1, and a perfect condition number would be 1. However, a condition number close to 1 indicates a well-conditioned matrix.

5. How can a matrix be improved to be better conditioned?

There are various techniques for improving the condition number of a matrix, such as scaling the matrix to reduce its largest singular value, using a different basis for the matrix, or using a different algorithm for solving the equations. However, these techniques may also affect the accuracy of the solutions, so it is important to find a balance between improving the condition number and maintaining accuracy.

Similar threads

  • Calculus and Beyond Homework Help
Replies
0
Views
41
  • Differential Equations
Replies
5
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
892
Replies
6
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
1K
Replies
5
Views
887
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
734
  • Quantum Physics
Replies
1
Views
665
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
1K
Back
Top