Matrix condition number question

In summary, the condition number of a matrix is a measure of its sensitivity to changes and is calculated by taking the ratio of the largest to the smallest singular value or using the eigenvalues. A higher condition number indicates an ill-conditioned matrix, leading to potential inaccuracies and instability in computations. It cannot be improved but can be reduced through numerical techniques for a more accurate and stable solution.
  • #1
tim51
3
0
Hey,

I've been studying condition numbers for matrices. I found a past exam question that asks if the notion of condition numbers can be used for non-square matrices.

Intuitively I thought it couldn't because cond(A) = ||A||.||A^-1|| and non-square matrices have no inverse. But MATLAB will calculate it?

Why do condition numbers exist for non-square matricies and do they have any meaning at all?

Thanks.
 
Physics news on Phys.org
  • #2
Least squares analysis involves non-square matrices and condition number is a meaure of rank deficiency.
 

1. What is the condition number of a matrix?

The condition number of a matrix is a measure of how sensitive the solution of a linear system is to changes in the matrix. It is the ratio of the largest to the smallest singular value of the matrix. A larger condition number indicates a more ill-conditioned matrix, meaning small changes in the matrix can result in large changes in the solution.

2. How is the condition number calculated?

The condition number of a matrix is calculated by taking the ratio of the largest singular value to the smallest singular value. It can also be calculated using the eigenvalues of the matrix.

3. What does a high condition number indicate?

A high condition number indicates that the matrix is ill-conditioned, meaning that small changes in the matrix can result in large changes in the solution. This can lead to inaccuracies and instability in numerical computations.

4. How does the condition number affect the accuracy of a solution?

The condition number is a measure of how sensitive the solution is to changes in the matrix. A higher condition number means that the solution is more sensitive to changes, leading to potential inaccuracies and instability. A lower condition number indicates a more well-conditioned matrix, with a more accurate and stable solution.

5. Can the condition number be improved?

The condition number of a matrix is a property that is inherent to the matrix and cannot be improved. However, by using numerical techniques such as pivoting or regularization, the condition number can be reduced to improve the accuracy and stability of the solution.

Similar threads

Prove the identity matrix is unique
    • Precalculus Mathematics Homework Help
2
Replies
57
Views
3K
A Question about factoring the Klein-Gordon equation
    • Quantum Physics
Replies
6
Views
1K
Proving statements about matrices | Linear Algebra
    • Calculus and Beyond Homework Help
Replies
25
Views
2K
MHB Can we simplify the relations of condition number for orthogonal matrices?
    • General Math
Replies
12
Views
3K
MHB Condition number - Relative error
    • General Math
Replies
9
Views
2K
MHB Is the Condition Number of A'A Related to its Matrix Norm?
    • General Math
Replies
3
Views
1K
MATLAB LMI in Matlab scaling to improve condition number?
    • MATLAB, Maple, Mathematica, LaTeX
Replies
2
Views
1K
B How can hyperreal numbers make infinitesimals logically sound in calculus?
    • Calculus
Replies
24
Views
3K
I Is a symmetric matrix with positive eigenvalues always real?
    • Linear and Abstract Algebra
Replies
8
Views
2K
Fast pentadiagonal matrix solver
    • Programming and Computer Science
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top