Lecture 37: More on diagonalization (Nicholson Section 5.5)
Sub Project: Linear Algebra Lecture Videos
Alternate Video Access via MyMedia | Video Duration: 48:40
Description: Started by reviewing the definition of what it means for a square matrix to be diagonalizable.
1:20 --- An nxn matrix A is diagonalizable if and only if you can find n linearly independent eigenvectors of A. Note: this lecture uses the language of linear independence, but Nicholson doesn’t get into that until Chapter 6. And so, in section 5.5 the book refers to “basic eigenvectors” and rather than asking that you have a full set of linearly independent eigenvectors, it asks that you have as many “basic eigenvectors” as the algebraic multiplicity of the eigenvalue. It’s the same thing.
5:30 --- Defined what it means for two square matrices to be “similar”. Reviewed a 3x3 example.
13:19 --- Stated theorem “Any set of eigenvectors with distinct eigenvalues is a linearly independent set”.
14:55 --- Compared the eigenvalues of A to the eigenvalues of the diagonal matrix. (They’re the same.) Compared the trace of A to the trace of the diagonal matrix. (They’re the same.) Compared the determinant of A to the determinant of the diagonal matrix. (They’re the same.)
18:40 --- returned to another prior 3x3 example.
25:45 --- Did a new 3x3 example. This one is super-important it’s a “nearly diagonal” matrix and it’s not diagonalizable.
34:14 --- Stated theorem that if A and B are similar matrices then they have the same eigenvalues, same determinant, same trace, and same rank. Note: this material isn’t in this section of Nicholson’s book. I provided it in because it’s important and beautiful and the proofs require you to understand some important things. But feel free to skip directly to 44:15.
36:00 -- I proved that the determinants are the same.
39:30 --- I proved that the eigenvalues are the same.
44:15 --- Did another 3x3 example. It’s not diagonalizable. But it’s nearly diagonalizable --- this is the Jordan Canonical Form theorem. It’s not in the course but it’s super-important and you’ll likely use it before you graduate.
