Eigenvalue Problem: Directional Loss & Consequences

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In summary, the concept of direction loses its ordinary meaning when applied to mathematical objects such as functions, harmonic modes, quantum states, and frequencies. Instead, it is given an abstract definition and is only used if it remains unchanged by a linear transformation. This is denoted by the prefix "eigen-" in terms such as eigenfunction, eigenmode, eigenstate, and eigenfrequency.
  • #1
jason.bourne
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Many kinds of mathematical objects can be treated as vectors: functions, harmonic modes, quantum states, and frequencies, for example. In these cases, the concept of direction loses its ordinary meaning, and is given an abstract definition. Even so, if this abstract direction is unchanged by a given linear transformation, the prefix "eigen" is used, as in eigenfunction, eigenmode, eigenstate, and eigenfrequency.

source: http://en.wikipedia.org/wiki/Eigenvalue



what are directional losses?
what are its consequences?
 
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  • #2
You are misreading the quoted text. There is nothing there about "directional losses", which I should say reminds me of an inefficient engine for some reason. The article says though vectors are often associated with direction, they may not always be and if those concepts which may be interpreted mathematically as directions remain untouched by a linear transformation, they are prefixed with "eigen-".
 
  • #3
"Direction", in the context of eigenvector and eigenvalues, does not always have its original intuative meaning. It's easy for everyone to understand that the vectors (1,2) and (3,4) have different "directions". You can draw a picture. It's less easy to accept that the functions(vectors) cos(x) and sin(x) have different "directions". Essentially we are overloading[/PLAIN] the english word "direction" to represent concepts far beyond its original scope.
 
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  • #4
okay. i got it.

thank you very much guys
 

Related to Eigenvalue Problem: Directional Loss & Consequences

1. What is an eigenvalue problem?

An eigenvalue problem is a mathematical problem that involves finding the values (known as eigenvalues) and corresponding vectors (known as eigenvectors) that satisfy a specific equation. It is commonly used in linear algebra and has applications in various fields, such as physics, engineering, and computer science.

2. How is the eigenvalue problem related to directional loss?

In the context of directional loss, the eigenvalue problem is used to determine the direction in which the loss is the most severe. The eigenvalue represents the amount of loss in that direction, while the corresponding eigenvector represents the direction itself.

3. What are the consequences of not solving the eigenvalue problem in directional loss analysis?

If the eigenvalue problem is not solved in directional loss analysis, it can lead to inaccurate or incomplete results. This could result in ineffective strategies for mitigating the directional loss, which could ultimately result in financial or operational consequences.

4. What factors can affect the results of the eigenvalue problem in directional loss analysis?

The results of the eigenvalue problem in directional loss analysis can be affected by various factors, such as the accuracy of the data used, the assumptions made in the analysis, and the complexity of the system being studied. It is essential to carefully consider and account for these factors to obtain reliable results.

5. How is the eigenvalue problem solved in directional loss analysis?

The eigenvalue problem is typically solved using numerical methods, such as the power method or the Jacobi method. These methods involve iterative processes that converge to the eigenvalues and eigenvectors of the system. It is also possible to solve the eigenvalue problem analytically for simpler systems with specific properties.

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