1. Eigenvalue/Eigenvector Definition
In the last lecture, we discussed scalar multiplication and its connection with matrices. Here we make the full connection via eigenvalues and eigenvectors by defining the two.
2. Definition
For a matrix A, the scalar lambda is an eigenvalue of A, with associated eigenvector v, if A times v equals lambda times v.
Notice what this is saying, it is saying that, for v, A has the same effect as a matrix multiplication on v as lambda does as a scalar multiplication on v. This is extremely important to grasp.
Notice that this can be any matrix A, it does not have to be a diagonal matrix like in the last lecture.
3. Example
Notice for A given above, multiplying A by the vector (1, 0) yields the vector (2, 0), which is simply 2 times the vector (1,0). Hence, 2 is an eigenvalue of A, with associated eigenvector (1,0). These two are often called an eigenpair of the matrix A.
4. Geometric Motivation
From a geometric perspective, an eigenvalue/eigenvector combination is essentially a combination that, whether the matrix represents a rotation, a reflection, a projection, or some combination of these elementary transformations, an eigenvector is a vector that stays fixed with respect to such an operation. It stays on the same line, so to speak.
The collection of eigenvectors for a matrix represent, in a sense, the characteristic dimensions of the matrix. A set of axes that stay invariant upon multiplication.
Notice that these axes needn't be the simple x and y axes. They can be tilted in some way.
5. Example, cont'd
Back to our last example, notice that not only is the vector (1, 0) an eigenvector associated with the eigenvalue 2, but so is (4, 0), the vector (1,0) multiplied by 4.
A times (1,0) is (2,0), the scalar 2 times (1,0), while A times (4,0) is (8,0), the scalar 2 times (4,0).
Hence if an eigenpair exists, that eigenpair can be altered by re-scaling the eigenvector to ones liking. Eigenvectors are, thus, entirely about direction and not magnitude. Scale is essentially problem-dependent.
6. Let's practice!
Time to explore some of these ideas in the exercises.