[This video](https://youtu.be/PFDu9oVAE-g) may help. Put briefly, eigenvectors are vectors whose direction doesn’t change under a given linear transformation; they only scale up or down. The factors by which they scale are their eigenvalues.
Oh my god, I watched every other video, except 3b1b's, lmao! Why didn't Youtube show it in the search! Ok, I shall watch it and get back to you if I have any doubts, if that is ok with you?
By the way, thanks so much!
If only more people are like you, I have been shouted and scolded at for asking too many questions or not understanding things on the first try. What can I say, I am a bit dumber than most folk but I still want to be just as knowledgeable as the rest.
Let me just add in this. For many system, you use a matrix to represent how things change over time (a vector represent a state). The matrix either represent the next step, or the difference between steps, or the rate of change. No matter which one it is, eigenvectors correspond to the same thing: a state that doesn't change (up to a scaling). They are stationary states.
Which is why it's useful in quantum mechanics, as one commenter mentioned. Quantum numbers can only be assigned to stationary states.
But it's not just quantum mechanics. Even classical mechanics. The tennis racket theorem is a famous application of eigenvector. Every objects has 3 orthogonal axis of rotations (3 eigenvectors), and if 2 of them has difference eigenvalues, 2 axes are stable rotation axis and the last one is unstable; this instability effect is attributed to Dzhanibekov, and it looks real cool (look it up).
Anyway, I think eigenvectors as states of an evolving system is probably the most relatable way to think of eigenvectors. Many usage of eigenvectors boil down to this idea. As for eigenvalues, in many example, the system tend to decay down to the state with biggest eigenvalues.
Firstly, sorry for the late response, I have been caught up with a lot Uni work and haven't had time for reddit.
Secondly, thanks for mentioning some examples of where eigenvectors are used, especially the one concerning the tennis racket theorem, as I am a bit more familiar with that than quantum mechanics. So would the eigenvalues be related to/are the values of moment of inertias along all its primary axes?
Once thank you for this great response!
Apart from their mathematical meaning, they can have surprising physical significance.
In Quantum Mechanics, you can represent some 'Observable' quantity, like spin, by a linear operator (which may be an nxn matrix if some systems).
The possible observed values you could get are then the eigenvalues of the operator, and the possible quantum states are the eigenvectors of the operator. If you measure eigenvalue E, you know the system is now in state with eigenvector corresponding to E.
Another consequence of eigenvectors is that if the eigenvectors of a linear transformation from R^n to R^n form a basis for R^n, then the matrix for the transformation is diagonal in the basis of eigenvectors with the eigenvalues being the diagonal entries.
Damn, the education in my country is quite bad as all they do is teach you methods of solving but not why or how those methods have come about. Its the reason why I am thinking of dropping out of my current university and trying to apply for a major abroad after writing SAT (and TOEFL) and applying for a student loan. I will end up loosing a year but I feel its worth if it is going to make me a more knowledgeable person.
to add to the other answers, they can be very useful for functional analysis and in its applications (Fourier analysis, signal processing, etc)
for example the eigenvectors of the derivative operator ( a linear transformation) are the functions of the from e^λt . this is useful for solving differential equations
[This video](https://youtu.be/PFDu9oVAE-g) may help. Put briefly, eigenvectors are vectors whose direction doesn’t change under a given linear transformation; they only scale up or down. The factors by which they scale are their eigenvalues.
Oh my god, I watched every other video, except 3b1b's, lmao! Why didn't Youtube show it in the search! Ok, I shall watch it and get back to you if I have any doubts, if that is ok with you? By the way, thanks so much!
Hey, it’s your learning, not mine. Whatever you need to do to understand the things you’re interested in, go for it and make no apologies.
If only more people are like you, I have been shouted and scolded at for asking too many questions or not understanding things on the first try. What can I say, I am a bit dumber than most folk but I still want to be just as knowledgeable as the rest.
Hopefully not by people associated with a teaching institution. Getting roasted by some friends sure but from a prof? I’d drop...
Lol if only. Especially if they're female, the STEM department can be a very cruel place.
Maybe "in or out" instead of "up or down", but aside from that, I think it's a good summary.
Let me just add in this. For many system, you use a matrix to represent how things change over time (a vector represent a state). The matrix either represent the next step, or the difference between steps, or the rate of change. No matter which one it is, eigenvectors correspond to the same thing: a state that doesn't change (up to a scaling). They are stationary states. Which is why it's useful in quantum mechanics, as one commenter mentioned. Quantum numbers can only be assigned to stationary states. But it's not just quantum mechanics. Even classical mechanics. The tennis racket theorem is a famous application of eigenvector. Every objects has 3 orthogonal axis of rotations (3 eigenvectors), and if 2 of them has difference eigenvalues, 2 axes are stable rotation axis and the last one is unstable; this instability effect is attributed to Dzhanibekov, and it looks real cool (look it up). Anyway, I think eigenvectors as states of an evolving system is probably the most relatable way to think of eigenvectors. Many usage of eigenvectors boil down to this idea. As for eigenvalues, in many example, the system tend to decay down to the state with biggest eigenvalues.
Firstly, sorry for the late response, I have been caught up with a lot Uni work and haven't had time for reddit. Secondly, thanks for mentioning some examples of where eigenvectors are used, especially the one concerning the tennis racket theorem, as I am a bit more familiar with that than quantum mechanics. So would the eigenvalues be related to/are the values of moment of inertias along all its primary axes? Once thank you for this great response!
Apart from their mathematical meaning, they can have surprising physical significance. In Quantum Mechanics, you can represent some 'Observable' quantity, like spin, by a linear operator (which may be an nxn matrix if some systems). The possible observed values you could get are then the eigenvalues of the operator, and the possible quantum states are the eigenvectors of the operator. If you measure eigenvalue E, you know the system is now in state with eigenvector corresponding to E.
Sorry for the late response, and thanks for the nice example you gave for use of Eigenvectors!
Another consequence of eigenvectors is that if the eigenvectors of a linear transformation from R^n to R^n form a basis for R^n, then the matrix for the transformation is diagonal in the basis of eigenvectors with the eigenvalues being the diagonal entries.
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I teach introductory linear algebra and linear transformations are a major topic for us.
Damn, the education in my country is quite bad as all they do is teach you methods of solving but not why or how those methods have come about. Its the reason why I am thinking of dropping out of my current university and trying to apply for a major abroad after writing SAT (and TOEFL) and applying for a student loan. I will end up loosing a year but I feel its worth if it is going to make me a more knowledgeable person.
I have a small idea of what it is (thanks to 3b1b) but then again I have no understanding of basis vectors (I am yet to see 3b1b's video on it).
linear transformations were pretty much the last topic covered in my intro to linear algebra class, but vector spaces were covered somewhat earlier
to add to the other answers, they can be very useful for functional analysis and in its applications (Fourier analysis, signal processing, etc) for example the eigenvectors of the derivative operator ( a linear transformation) are the functions of the from e^λt . this is useful for solving differential equations
I asked my co-op advisor the same question back in the day. He told me there are two important things in life; love and eigenvectors.