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jason.bourne
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source: http://en.wikipedia.org/wiki/Eigenvalue
what are directional losses?
what are its consequences?
Eigen function, eigen value, eigen vector
- Thread starter jason.bourne
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In summary, vectors can be defined as a row of numbers with an index to indicate the column, and this concept can also be applied to continuous functions. In such cases, the idea of direction may not have its usual meaning, but it can still be thought of in an abstract sense. If this abstract direction remains unchanged under a linear transformation, the prefix "eigen" is used to refer to it, as seen in terms like eigenfunction, eigenmode, eigenstate, and eigenfrequency. The analogy of direction in "real" space and "function" space can be useful in understanding this concept.
source: http://en.wikipedia.org/wiki/Eigenvalue
what are directional losses?
what are its consequences?
- #2
atyy
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Roughly speaking, a vector is just an ordered row of numbers, where you can use an index i to indicate what column a number is in.
That's pretty much what a continuous function f(x) is also, except that you now use a continuous index x to indicate what "column" a number is in.
So you can treat a continuous function like a vector (ie. you can add it, take scalar products etc.). We think of vectors as things having direction. Since functions can be thought of like vectors, we can use that analogy to think that functions also have some sort of "direction". This is just an analogy, which is why the author of the article you quoted said "the concept of direction loses its ordinary meaning". There aren't really any consequences, because actually the analogy works in great detail and can be usefully exploited.
I would actually say it differently: A normal vector gives us the direction in "real" space. Thinking of the function as a vector gives us a sense of direction in "function" space.
There are some differences between finite and infinite dimensional "vector" spaces, but the above is a quick and dirty way to think about it.
That's pretty much what a continuous function f(x) is also, except that you now use a continuous index x to indicate what "column" a number is in.
So you can treat a continuous function like a vector (ie. you can add it, take scalar products etc.). We think of vectors as things having direction. Since functions can be thought of like vectors, we can use that analogy to think that functions also have some sort of "direction". This is just an analogy, which is why the author of the article you quoted said "the concept of direction loses its ordinary meaning". There aren't really any consequences, because actually the analogy works in great detail and can be usefully exploited.
I would actually say it differently: A normal vector gives us the direction in "real" space. Thinking of the function as a vector gives us a sense of direction in "function" space.
There are some differences between finite and infinite dimensional "vector" spaces, but the above is a quick and dirty way to think about it.
- #3
jason.bourne
- 82
- 1
yeah i get it.
thank you so much
thank you so much
What is an eigenfunction?
An eigenfunction is a specific type of function that, when multiplied by a constant factor, remains unchanged. In other words, the function is only scaled, but its shape and form remain the same. Eigenfunctions are commonly used in mathematical and scientific fields, such as physics and engineering.
What is an eigenvalue?
An eigenvalue is a number that represents the scaling factor of an eigenfunction. It is the constant by which the eigenfunction is multiplied to remain unchanged. In other words, it is the value that makes the eigenfunction an eigenfunction.
What is an eigenvector?
An eigenvector is a vector that, when multiplied by a transformation matrix, is only scaled by a constant factor. In other words, the eigenvector is only stretched or shrunk, but its direction remains unchanged. Eigenvectors are commonly used in linear algebra and are closely related to eigenfunctions and eigenvalues.
What is the significance of eigenfunctions, eigenvalues, and eigenvectors?
Eigenfunctions, eigenvalues, and eigenvectors are important concepts in mathematics and science because they allow us to study and understand complex systems in a simplified manner. They are used in a variety of fields, such as quantum mechanics, signal processing, and data analysis, to solve problems and make predictions. Eigenfunctions, eigenvalues, and eigenvectors also have many practical applications, such as in image and audio compression algorithms.
How are eigenfunctions, eigenvalues, and eigenvectors calculated?
The process of finding eigenfunctions, eigenvalues, and eigenvectors involves solving a specific mathematical equation called an eigenvalue problem. This can be done using various methods and techniques, such as the power method, Jacobi method, or the QR algorithm. In some cases, eigenfunctions, eigenvalues, and eigenvectors can also be found analytically using formulas and equations specific to the problem at hand.
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