Question 1

What are the pre-requisites for this lesson? How do I determine what other videos I need to watch in order to understand this one?

I have < 1 week (for a Quantum Computing course), it mentions specifically this and one other Linear Algebra topic ("orthogonal bases"). I've been serially watching every video in the "Linear Algebra" section from the beginning, but there will not be enough time.

So, how to determine what videos I can skip in order to reach this one and be able to understand it?

Accepted Answer

You should know a fair amount of linA for this lesson. So Gauss-Jordan, rank, null space, orthogonal, et cetera. While all of these things may not come up, they will give you a better understanding of what is doing on.

Question 2

So if an eigenvector is a vector transformed from an original vector and an eigenvalue is the scaler multiplier, why do we give them those fancy names anyway? Is it because those values and vectors will produce a perfect base or something instead of searching randomly for a perfect base or a value to transform a set of vectors according to our needs? Why do we even came up with these as humans?

Accepted Answer

Think of a matrix as if it were one, big, complex number.  One of the rules of integers says the prime factorization of any integer is unique... if two numbers have the same prime factorization then they must be the same number.

It becomes useful (later) to be able to say if two matrices are the same, but it can be very difficult (because the matrices are huge, their values are complex or worse... a variety of causes).  Eigenvalues are one part of a process that leads (among other places) to a process analogous to prime factorization of a matrix, turning it into a product of other matrices that each have a set of well-defined properties.  Those matrices are tolerably easy to produce, and if two matrices can be 'factored' into the same sets of matrix products, then they are 'equal'.

And that's just up to the first half of grad school :)

Question 3

I think I'm realising why I have found this topic slippery in the past. Some presentations (like this one) emphasise the transformational role of the matrix we're finding eigenvectors/values for, i.e. it acts upon vectors to make new vectors; others speak of it more as a static dataset to be analyzed, i.e. a collection of existing datapoints-as-vectors. Bridging those two views is quite subtle... How should we come to think of a dataset as a transformation?

Accepted Answer

"How should we come to think of a dataset as a transformation?"  By comparing the dataset you're mapping into with the dataset you're mapping from. If both datasets are exactly the same, the transformation is unitary, i.e, the eigenvector is just a set of 1's. 
If the dataset you're mapping into is different from the dataset you've mapped from, then you can often (but not always) find some linear function that relates them. Say, for example, that the rows in the dataset you're mapping into are each multiplied by 2 while the columns are each multiplied by 3. That's a linear transformation, and it can be captured in a matrix that gives an eigenvector that performs that transformation,  AKA a linear mapping.

Question 4

what are the applications of eigen values and eigen vectors in real life?...i need a simple answer. i read about tacoma bridge but could not really understand it...

Accepted Answer

Eigenvalues and eigenvectors prove enormously useful in linear mapping. Let's take an example: suppose you want to change the perspective of a painting. If you scale the x direction to a different value than the y direction (say x -> 3x while y -> 2y), you simulate a change of perspective. This would represent what happens if you look a a scene from close to the ground as opposed to higher in the air.  Objects appear distorted and foreshortened. A change in perspective in a painting is really just a vector transformation that performs a linear map. That is, a set of points (the painting) gets transformed by multiplying the x distance by one value and the y value by a different value. You can capture the process of doing this in a matrix, and that matrix represents a vector that's called the eigenvector.   
If the mapping isn't linear, we're out of the realm of the eigenvector and into the realm of the tensor. So eigenvectors do well with linear mappings, but not with nonlinear mappings. In the case of nonlinear mappings, the fixed points in the eigenvector matrix would be replaced with functions that can take on many different values.  
Eigenvectors pop up in the study of the spread of infectious diseases or vibration studies or heat transfer because these are generally linear functions. Diseases tend to spread slowly, heat spreads gradually, and vibrations propagate gradually.  Diseases and vibrations and heat patterns also tend to spread by contact, so you don't get oddball discontinuous patterns of disease spread of vibration propagation or heat transfer (typically). This means that heat patterns and vibration propagation and disease spreading can be simulated reasonably well by linear maps from one set of points to another set of points. 
Nonlinear discontinuous systems like explosions or the orbits of a 3-body system in a gravitational field or a star  undergoing gravitational collapse don't lend themselves to simple linear maps. As a result, eigenvectors do not offer a good way of describing those systems. For those kinds of nonlinear systems, you typically need tensors instead of linear maps.

Question 5

can anyone provide data and videos relating to the daily life use of eigenvalues and eigenvectors i.e physical significance or visualization?

Accepted Answer

Visual perspective. Imagine looking at your own legs while you're lying down in bed. Your legs seem to be foreshortened. Now imagine looking at your legs while you're standing in front of a mirror. Your legs appear to be much longer. 
The difference in these two views is captured by a linear transformation that maps one view into another. This linear transformation gets described by a matrix called the eigenvector. The points in that matrix are called eigenvalues.
Think of it this way: the eigenmatrix contains a set of values for stretching or shrinking your legs. Those stretching or shrinking values are eigenvalues. The eigenvector contins a set of directions for stretching or shrinking your legs. Those stretching or shrinking values are eigenvectors.
These kinds of linear transformations prove absolutely vital in doing CGI animation in movies.  The eigenmatrices and eigenvectors change as you change the location of the virtual camera in a CGI animation.
Eigenvectors and eigenvalues are also vital in interpreting data from a CAT scan. In that case you have a set of X-ray values and you want to turn them into a visual scene. But you don't just have one set of X-ray scans, you do a bunch of X-ray scans in layers, like layers in a cake. From all these layers of X-ray scans, you want to build up a picture of a 3-Dimensional object, a part of the human body, and you want to be able to rotate that 3-D image in the computer so you can see it from different viewpoints. This requires a set of linear mappings, and in turns demands eigenvalues and eigenvectors.

Question 6

I have a question, if we have some vectors that are multiples of each other i.e. they have the same line span, does this mean that they have the same eiegenvalues ?

Accepted Answer

Yes, say *v* is an eigenvector of a matrix A with eigenvalue λ. Then Av=λv. Let's verify c*v (where c is non zero) is also an eigenvector of eigenvalue λ. You can verify this by computing A(cv)=c(Av)=c(λv)=λ(cv). Thus cv is also an eigenvector with eigenvalue λ. I wrote c as non zero, because eigenvectors are non zero, so c*v cannot be zero.

Linear algebra

Course: Linear algebra > Unit 3

Introduction to eigenvalues and eigenvectors

Want to join the conversation?

Video transcript