1. Solving Eigenvalue/Eigenvector Problems
While it's always important to determine if a given candidate is a solution for a problem of interest, it is, of course, the most important skill to be able to identify candidate solutions.
We will do this in this lecture.
2. Properties of Solutions to Eigenvalue/Eigenvector Problems
Before we compute some solutions to eigenvalue/eigenvector problems, we discuss some properties of such problems:
First, an n by n matrix has n eigenvalues, up to multiplicity. This means that a 3x3 matrix can have up to three distinct eigenvalues, but if two eigenvalues repeat, there may only be two, for example.
Second, eigenvalues need not be real. A matrix will all real entries very well could have complex or completely imaginary eigenvalues. This fact has some interesting upshots in areas like differential equations, where they are the impetus for defining trig functions like sine and cosine. Complex eigenvalues, however, must come in conjugate pairs. So if 1 + 2i is an eigenvalue of a matrix, 1 - 2i must also be one.
3. Solving Eigenvalue/Eigenvector Problems in R
When it comes to actually solving these problems in R, the code is simple, eigen (with lowercase letters).
The output can be a little to unpack, so we first simply call the command and see what it spits out.
4. Solving Eigenvalue/Eigenvector Problems in R
Notice here that you can store the output of the eigen function as a list, in this case E. E has values, called with $ values and vectors, called with $ vectors.
You can extract individual eigenvalues by using the index, in this case 1, to extract the first eigenvalue. The eigenvalues are in descending order by their modulus (absolute value).
You can extract individual eigenvectors by using the index as well. Since this is a column, you have to use the second index. These vectors are indexed by which eigenvalue they are associated with.
5. Example with Complex Eigenvalues
Here's an example of a two-by-two matrix whose eigenvalues, and associated eigenvectors, are both complex.
Notice that both eigenvalues are complex conjugates of each other, as our theory states. If multiplied together, they make a real number.
6. Let's practice
Let's explore these topics in the exercises.