In this lecture, if we're on schedule, we will be able to start work on eigen values and eigen vectors.
In doing this we'll be skipping right over chapters 4 and 5; but the ideas of eigen vectors and eigen values are maybe the nicest in the course, so we want to be able to spend enough time on them. We will come back to chapters 4 and 5 later.
We mentioned earlier that it's much easier to multiply diagonal matrices. That means it's much easier to find what happens when a transformation with a diagonal matrix is repeated.
What is the fourth power of
| / | 2 | 0 | \ |
| \ | 0 | 1 | / |
We can quickly see the effect this transformation on a graph:
how about finding the fourth power of
| / | 3/2 | 1/2 | \ |
| \ | 1/2 | 3/2 | / |
or what about
| / | 2 | -1 | \ |
| \ | 0 | 1 | / |
Below is the effect of
| / | 3/2 | 1/2 | \ |
| \ | 1/2 | 3/2 | / |
I've drawn the effect of sucessive applications of the transformaion on the right. I've used the way the standard grid is transformed to work out where the succesive images must be. This is done by looking at the coordinates relative to the original grid; the transformed coordinates must have the same relation to the transformed grid. I'll explain this in class.
For the above graph, we can ask the following questions:
There are two directions which define a grid, which is transformed in a simple way:
How could we have found this grid without drawing the pictures?
If we could do that for other matices, we'd be able to see the effect of repeated transformations quickly. This is what diagonalisation is about.
The directions which define the grid are the eigen vectors, and the amounts of stretch in these directions are called the eigen values. So for
| / | 3/2 | 1/2 | \ |
| \ | 1/2 | 3/2 | / |
For
| / | 2 | 0 | \ |
| \ | 0 | 1 | / |
There are a lot of problems were we want to find powers of matrices, so the eigen vectors and eigen values are very useful.
We will learn about how to find the eigen values and eigen vectors.
There will be lots of applications of these ideas.
Eigen Vectors
An eigen vector of a matrix A is a vector v such that A sends v to a multiple of itself. The multiple is called the eigen value corresponding to that vector. If the eigen value is L, we have
If we know what an eigen value is, it's pretty easy to find what the corresponding eigen vector is. We did this in class, and it's covered in section 6.1 of the notes. All you have to do is find a non trivial solution of the equation
If A is an upper triangular matrix, then the eigen values must be the numbers occuring on the diagonal (see end of section 6.1).
How can we find the eigen values if the matrix is not upper triangular?
If A isn't upper triangular, we can start off in the same way, and we'll end up seeing that there is a polynomial called the characteristic polynomial, and it's roots are the eigen values. This polynomial is found using the determinant, which we'll also look at.
We will cover the population movement example again, from section 2.7,
with a more sophisticated calculation now, in example 5 of section 6.2.
Diagonalization
Above, we saw what it means to find the diagonalization of a matrix, for an example. In general, to diagonalize a matrix A means that there is a matrix P, and a diagonal matrix D, with
| / | a | 0 | 0 | \ |
| | | 0 | b | 0 | | |
| \ | 0 | 0 | c | / |
Now we'll see how this helps us to find powers of matrices, and we'll see a method of finding the matrix P that diagonalizes A.
If
D8 is easy to find, so now we can find A8 easily too.
We will talk a little about what diagonalization means geometrically, with reference to the above example.
Please send comments, questions, suggestions, corrections, ideas, to me.
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