Lecture 23: How to find eigenvalues and eigenvectors (Nicholson Section 3.3)
Sub Project: Linear Algebra Lecture Videos
Alternate Video Access via MyMedia | Video Duration: 49:24
Description: Note: In this lecture, I use the language of “linear combinations” --- you may not have seen this language yet but I hope it’s clear enough what is meant. Reviewed definition of eigenvector and eigenvalue. Reviewed why we’re looking for lambdas so that det(A-lambda I) = 0.
6:30 --- Returned to the reflection example from previous lecture. We know the eigenvalues & eigenvectors geometrically but how could we have found them algebraically? Worked through the example.
16:45 --- Important! What happens if you’d made a mistake when you computed your eigenvalues? What happens when you then try to find eigenvectors?
21:30 --- Did a 3x3 example. Here we don’t have geometric intuition and we’re going to have to compute the eigenvalues by finding the roots of a cubic polynomial. This example’s interesting because we get a repeated eigenvalue and so when we look for eigenvectors we get them in two different directions.
44:10 --- Important! If you add two eigenvectors together and they have different eigenvalues, is the sum also an eigenvector? No!
46:15 --- Slammed through a final 3x3 example, introduced the language of “algebraic multiplicity” of eigenvalues. In this example, there was a repeated eigenvalue but I couldn’t find two eigenvectors w/ different directions.
