Min eigenvalue of a perturbed matrix

In summary, the conversation revolved around a problem posted in the 'Linear and abstract algebra' section, which was related to matrices. The person who posted the problem, a physicist, acknowledged that it may involve more analysis than algebra. They also mentioned a previous question that was difficult to read and received a suggestion to use LaTeX. The person thanked the responder and stated their intention to start a new thread with clearer information.
  • #1
julian
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I posted a problem called "estimating eigenvalue of perturbed matrix" in the section 'Linear and abstract algebra' cus, well it was to do with matrices (I'm a physicist - appologies for that). Actually come to think of it the problem probably has more to do with analysis...If a kind maths peep could have a look I'd be grateful.
 
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  • #2
Your other question is pretty hard to read. You'll probably get more responses if you use the LaTeX
 
  • #3
I see how you use LaTex on the forum now - didn't know about the itex command. Thanks. Shall just start the thread again and try to be a bit clearer.
 

1. What is the definition of the minimum eigenvalue of a perturbed matrix?

The minimum eigenvalue of a perturbed matrix is the smallest eigenvalue of the perturbed matrix, which is a matrix that has been slightly modified from the original matrix.

2. How is the minimum eigenvalue of a perturbed matrix calculated?

The minimum eigenvalue of a perturbed matrix can be calculated by finding the eigenvalues of the perturbed matrix and then selecting the smallest one. This can be done using numerical methods such as the power iteration method or the QR algorithm.

3. What does the minimum eigenvalue of a perturbed matrix represent?

The minimum eigenvalue of a perturbed matrix represents the smallest possible value that can be obtained by multiplying the matrix by a non-zero vector. It is an important measure of the stability and behavior of a system described by the matrix.

4. How does the minimum eigenvalue of a perturbed matrix relate to the condition number?

The minimum eigenvalue of a perturbed matrix is inversely proportional to the condition number of the matrix. This means that a lower minimum eigenvalue indicates a higher condition number, which in turn indicates a less stable and well-conditioned system.

5. How can the minimum eigenvalue of a perturbed matrix be used in practical applications?

The minimum eigenvalue of a perturbed matrix can be used to analyze the stability and behavior of systems in diverse fields such as engineering, physics, and economics. It can also be used in numerical computations to determine the accuracy and reliability of results.

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