ELI5- Eigenvalues and Eigenvectors?
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Lots of good explanations here. Glad I stumbled on this page
I found [this](https://www.mathsisfun.com/algebra/eigenvalue.html) explanation to be very understandable. The examples with real numbers really help.
Lots of good explanations here. Glad I stumbled on this page
I found [this](https://www.mathsisfun.com/algebra/eigenvalue.html) explanation to be very understandable. The examples with real numbers really help.
Suppose I take a rubber sheet and stretch it to twice its length vertically, and squeeze it so it’s half its length horizontally. Are there any shapes I can draw on the sheet that would keep the same shape and orientation when I do this, and just get bigger or smaller?
Yes. Vertical and horizontal lines will keep their shape when stretched. Plus a point at the center, but that’s boring. For other kinds of stretches, the lines that keep their shape may be diagonals, but there’s usually two for a rubber sheet.
The directions of constant-shape stretches are the “eigenvectors”, the amount of stretch in those directions are the “eigenvalues”.
This is useful because once we know how points on these lines behave when stretching, we can figure out what happens to *any other point* pretty easily. In my simple horizontal/vertical example, we just have to figure out how far any point is from the center horizontally and vertically, and double or halve that. For other kinds of stretches, we might have to measure diagonal distances.
But nobody cares about rubber sheets. But in many fields, we often deal with physical and mathematical systems that transform quantities the same way this rubber sheet does. One combination of quantities gets twice as big while another combination gets half as big. Eigenvectors are useful for finding simple solutions to complicated “stretching” problems like this, whether the solutions involve 2 variables (equivalent to stretching a 2-d rubber sheet), 3 (stretching a 3-d rubber block), or millions.
And for extra credit, what about rotations? Suppose I *turn* my rubber sheet instead of stretching it, are there any shapes that keep their shape and orientation? Surprisingly the answer is “yes”, if we’re willing to consider eigenvectors and eigenvalues with imaginary numbers in them! Now that probably sounds like mathematical nonsense, but if you play video games, your computer graphics card is solving this sort of mathematical nonsense a trillion times a second.
Suppose I take a rubber sheet and stretch it to twice its length vertically, and squeeze it so it’s half its length horizontally. Are there any shapes I can draw on the sheet that would keep the same shape and orientation when I do this, and just get bigger or smaller?
Yes. Vertical and horizontal lines will keep their shape when stretched. Plus a point at the center, but that’s boring. For other kinds of stretches, the lines that keep their shape may be diagonals, but there’s usually two for a rubber sheet.
The directions of constant-shape stretches are the “eigenvectors”, the amount of stretch in those directions are the “eigenvalues”.
This is useful because once we know how points on these lines behave when stretching, we can figure out what happens to *any other point* pretty easily. In my simple horizontal/vertical example, we just have to figure out how far any point is from the center horizontally and vertically, and double or halve that. For other kinds of stretches, we might have to measure diagonal distances.
But nobody cares about rubber sheets. But in many fields, we often deal with physical and mathematical systems that transform quantities the same way this rubber sheet does. One combination of quantities gets twice as big while another combination gets half as big. Eigenvectors are useful for finding simple solutions to complicated “stretching” problems like this, whether the solutions involve 2 variables (equivalent to stretching a 2-d rubber sheet), 3 (stretching a 3-d rubber block), or millions.
And for extra credit, what about rotations? Suppose I *turn* my rubber sheet instead of stretching it, are there any shapes that keep their shape and orientation? Surprisingly the answer is “yes”, if we’re willing to consider eigenvectors and eigenvalues with imaginary numbers in them! Now that probably sounds like mathematical nonsense, but if you play video games, your computer graphics card is solving this sort of mathematical nonsense a trillion times a second.
Lots of good explanations here. Glad I stumbled on this page
Suppose I take a rubber sheet and stretch it to twice its length vertically, and squeeze it so it’s half its length horizontally. Are there any shapes I can draw on the sheet that would keep the same shape and orientation when I do this, and just get bigger or smaller?
Yes. Vertical and horizontal lines will keep their shape when stretched. Plus a point at the center, but that’s boring. For other kinds of stretches, the lines that keep their shape may be diagonals, but there’s usually two for a rubber sheet.
The directions of constant-shape stretches are the “eigenvectors”, the amount of stretch in those directions are the “eigenvalues”.
This is useful because once we know how points on these lines behave when stretching, we can figure out what happens to *any other point* pretty easily. In my simple horizontal/vertical example, we just have to figure out how far any point is from the center horizontally and vertically, and double or halve that. For other kinds of stretches, we might have to measure diagonal distances.
But nobody cares about rubber sheets. But in many fields, we often deal with physical and mathematical systems that transform quantities the same way this rubber sheet does. One combination of quantities gets twice as big while another combination gets half as big. Eigenvectors are useful for finding simple solutions to complicated “stretching” problems like this, whether the solutions involve 2 variables (equivalent to stretching a 2-d rubber sheet), 3 (stretching a 3-d rubber block), or millions.
And for extra credit, what about rotations? Suppose I *turn* my rubber sheet instead of stretching it, are there any shapes that keep their shape and orientation? Surprisingly the answer is “yes”, if we’re willing to consider eigenvectors and eigenvalues with imaginary numbers in them! Now that probably sounds like mathematical nonsense, but if you play video games, your computer graphics card is solving this sort of mathematical nonsense a trillion times a second.
Imagine you have a big, magical toy box with lots of different toys inside. Each toy has a special power or ability, like being able to jump really high or change colors. Now, let’s say you have a special toy in your box that can transform any other toy into a new toy with a different power.
This special toy is like an “operator” in math, and the toys it transforms are like “vectors”. When the operator transforms a vector, it can give it a new power or ability, just like the special toy in your toy box.
Now, let’s say that there are certain vectors that, when transformed by the operator, end up becoming multiples of themselves. These special vectors are called “eigenvectors”, and the multiples that they become are called “eigenvalues”.
Think of it like this: if you have a toy car that can transform into a toy plane, and you transform it using the special toy in your toy box, it might become a toy plane that is twice as big as the original car. In this case, the car is the eigenvector and the factor of two is the eigenvalue.
So, in math, eigenvectors and eigenvalues are special properties of operators and vectors that can help us understand how they behave and change. They’re like special toys and powers that can help us solve problems and do cool things!
Imagine you have a big, magical toy box with lots of different toys inside. Each toy has a special power or ability, like being able to jump really high or change colors. Now, let’s say you have a special toy in your box that can transform any other toy into a new toy with a different power.
This special toy is like an “operator” in math, and the toys it transforms are like “vectors”. When the operator transforms a vector, it can give it a new power or ability, just like the special toy in your toy box.
Now, let’s say that there are certain vectors that, when transformed by the operator, end up becoming multiples of themselves. These special vectors are called “eigenvectors”, and the multiples that they become are called “eigenvalues”.
Think of it like this: if you have a toy car that can transform into a toy plane, and you transform it using the special toy in your toy box, it might become a toy plane that is twice as big as the original car. In this case, the car is the eigenvector and the factor of two is the eigenvalue.
So, in math, eigenvectors and eigenvalues are special properties of operators and vectors that can help us understand how they behave and change. They’re like special toys and powers that can help us solve problems and do cool things!
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