Compute Condition Number of Matrix A

  • Thread starter DeadxBunny
  • Start date
  • Tags
    Condition
In summary, the condition number of the given matrix, A, is equal to 2-2E, where E is much smaller than the l infinity norm. The norm of A is found to be 2, while the norm of A^-1 is 1-E. The calculation for A^-1 is incorrect and the resulting condition number is incorrect as well. It is evident that the condition number significantly increases as E approaches zero.
  • #1
DeadxBunny
30
0

Homework Statement


Compute the condition number of the following matrix:

| 1 1 |
| E -E |

where E<<l infinity norm.


Homework Equations



l infinity norm: ||x||infinity = max |xi| where i goes from 1 to n.

Condition number: K(A) = ||A||*||A^-1||



The Attempt at a Solution


A=
| 1 1 |
| E -E |

A^-1=
| 1 -E |
| -1 -E |

||A||infinity = max(2,0) = 2
||A^-1||infinity = max(1-E,-1-E) = 1-E

K(A) = 2(1-E) = 2-2E

Is this right or am I completely off? Thanks!
 
Physics news on Phys.org
  • #2
Your result for A^-1 is wrong.
Additionally, you gave the definition for the norm of a vector but explained nothing concerning the norm of matrix. You should also explain how you calculated the norm of A and A^-1.
The end result is wrong of course.
It should be clear that the condition number increases a lot when E goes to zero.
 

FAQ: Compute Condition Number of Matrix A

What is the condition number of a matrix?

The condition number of a matrix A is a measure of how sensitive the solution of a linear system Ax = b is to changes in the matrix A. It is denoted by cond(A) and can be thought of as the ratio of the maximum and minimum singular values of A.

How is the condition number of a matrix computed?

The condition number of a matrix can be computed using various methods, such as the singular value decomposition (SVD) or the eigenvalue decomposition. One common method is to use the norm of the matrix, where the condition number is equal to the norm of A multiplied by the norm of the inverse of A.

What does a high condition number indicate?

A high condition number indicates that the matrix A is ill-conditioned, meaning that small changes in the input can result in large changes in the output. In other words, the matrix is close to being singular, making it difficult to obtain an accurate solution for the linear system Ax = b.

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

The condition number of a matrix is directly related to the accuracy of the solution of a linear system. The higher the condition number, the less accurate the solution will be. This is because small changes in the input can result in large changes in the output, making it difficult to obtain a precise solution.

Can the condition number of a matrix be reduced?

The condition number of a matrix cannot be reduced, as it is an inherent property of the matrix itself. However, techniques such as regularization or using a different matrix representation can help improve the accuracy of the solution for a high condition number matrix.

Similar threads

Solving a first order matrix differential equation
Replies
6
Views
834
I cannot make the Gram-Schmidt procedure work
Replies
16
Views
2K
Trouble with Matrix Exponentials
Replies
4
Views
943
Solving the Density of States: Understanding dn/dE
Replies
7
Views
1K
Understanding Conditional Expectation, Variance, and Precision Matrices
Replies
0
Views
560
Given specific v, dimension of subspace of L(V, W) where Tv=0?
Replies
15
Views
2K
Linear homogenous system with repeated eigenvalues
Replies
2
Views
1K
Finding convergence of this series using Integral/Comparison
Replies
2
Views
1K
How do I get the solution with the matrix exponential method
Replies
11
Views
2K
Fourier transform of integral e^-a|x|
Replies
6
Views
2K

Hot Threads

Recent Insights

Back
Top