ELI5- Eigenvalues and Eigenvectors?
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Well the premise itself is pretty simple. An eigenvector is like one side of an equation, and the eigenvalue is some value that makes the overall equation true when it is multiplied by the other side. Of course this usually involve matrices which makes it a bit more complicated, but that is the general idea. This theroem has many implications depending on your field, but that stuff is much more than eli5.
Scaling things up and down is very easy to work with and think about. It’s just zooming in and out. But if you can only zoom in and out, then you don’t have too much to work with. Luckily, we can zoom different directions in different ways. So I could vertically zoom in by a factor of 3 and horizontally zoom in by a factor of 10. This is kinda like a “mixed zoom”. (This would be the transformation (x,y)->(10x,3y).)
Eigen-stuff is just figuring out “mixed zooms”. To have a mixed zoom, you need the directions that you’re zooming in (the “eigenvectors”) and the factors by which you’re zooming (the “eigenvalues”). Important theorems in math tell us that *most* nice spacial transformations are actually “mixed zooms”, and eigenvector and eigenvalue methods are how to find these directions and scale factors. This, overall, simplifies work because we don’t have to deal with crazy, high dimensional transformations and can just think about different zooms in different directions.
Scaling things up and down is very easy to work with and think about. It’s just zooming in and out. But if you can only zoom in and out, then you don’t have too much to work with. Luckily, we can zoom different directions in different ways. So I could vertically zoom in by a factor of 3 and horizontally zoom in by a factor of 10. This is kinda like a “mixed zoom”. (This would be the transformation (x,y)->(10x,3y).)
Eigen-stuff is just figuring out “mixed zooms”. To have a mixed zoom, you need the directions that you’re zooming in (the “eigenvectors”) and the factors by which you’re zooming (the “eigenvalues”). Important theorems in math tell us that *most* nice spacial transformations are actually “mixed zooms”, and eigenvector and eigenvalue methods are how to find these directions and scale factors. This, overall, simplifies work because we don’t have to deal with crazy, high dimensional transformations and can just think about different zooms in different directions.
Scaling things up and down is very easy to work with and think about. It’s just zooming in and out. But if you can only zoom in and out, then you don’t have too much to work with. Luckily, we can zoom different directions in different ways. So I could vertically zoom in by a factor of 3 and horizontally zoom in by a factor of 10. This is kinda like a “mixed zoom”. (This would be the transformation (x,y)->(10x,3y).)
Eigen-stuff is just figuring out “mixed zooms”. To have a mixed zoom, you need the directions that you’re zooming in (the “eigenvectors”) and the factors by which you’re zooming (the “eigenvalues”). Important theorems in math tell us that *most* nice spacial transformations are actually “mixed zooms”, and eigenvector and eigenvalue methods are how to find these directions and scale factors. This, overall, simplifies work because we don’t have to deal with crazy, high dimensional transformations and can just think about different zooms in different directions.
Well the premise itself is pretty simple. An eigenvector is like one side of an equation, and the eigenvalue is some value that makes the overall equation true when it is multiplied by the other side. Of course this usually involve matrices which makes it a bit more complicated, but that is the general idea. This theroem has many implications depending on your field, but that stuff is much more than eli5.
Well the premise itself is pretty simple. An eigenvector is like one side of an equation, and the eigenvalue is some value that makes the overall equation true when it is multiplied by the other side. Of course this usually involve matrices which makes it a bit more complicated, but that is the general idea. This theroem has many implications depending on your field, but that stuff is much more than eli5.
I’m still looking for a good ELI5, but this article at least makes it a bit more understandable: [https://wiki.pathmind.com/eigenvector](https://wiki.pathmind.com/eigenvector)
>So out of all the vectors affected by a matrix blowing through one space, which one is the eigenvector? It’s the one that that changes length but not direction; that is, the eigenvector is already pointing in the same direction that the matrix is pushing all vectors toward. An eigenvector is like a weathervane. An eigenvane, as it were.
I’m still looking for a good ELI5, but this article at least makes it a bit more understandable: [https://wiki.pathmind.com/eigenvector](https://wiki.pathmind.com/eigenvector)
>So out of all the vectors affected by a matrix blowing through one space, which one is the eigenvector? It’s the one that that changes length but not direction; that is, the eigenvector is already pointing in the same direction that the matrix is pushing all vectors toward. An eigenvector is like a weathervane. An eigenvane, as it were.
I’m still looking for a good ELI5, but this article at least makes it a bit more understandable: [https://wiki.pathmind.com/eigenvector](https://wiki.pathmind.com/eigenvector)
>So out of all the vectors affected by a matrix blowing through one space, which one is the eigenvector? It’s the one that that changes length but not direction; that is, the eigenvector is already pointing in the same direction that the matrix is pushing all vectors toward. An eigenvector is like a weathervane. An eigenvane, as it were.
I found [this](https://www.mathsisfun.com/algebra/eigenvalue.html) explanation to be very understandable. The examples with real numbers really help.
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