Linear Algebra - Lecture 39: The Characteristic Polynomial and Multiplicity

We talk about the characteristic polynomial of a matrix, and what polynomials can tell us about eigenvalues. We also introduce the algebraic and geometric multiplicities of an eigenvalue, and we talk about why the eigenvalues of a triangular matrix are its diagonal entries.

Please leave a comment below if you have any questions, comments, or corrections.

Timestamps:
00:00 - Introduction and definition
03:08 - 3x3 example
09:47 - Algebraic and geometric multiplicities
15:20 - The Fundamental Theorem of Algebra (sum of algebraic multiplicities)
20:25 - Eigenvalues of triangular matrices

#linearalgebra #eigenvaluesandeigenvectors #matrices #math

[pgfkekj]

Thank you! you explained it very well in simple language, I will pass in my exam because of you!

[AG-cfet]

In which cases does the alg multiplicity differ from geometric and why? In the cases where the quadratic nxn Matrix does transform on a lower dimensional subspace (det=0) it‘s clear, the Eigenvectors can not span the whole n dim space but in this example at the beginning even though we have full rank, the eigenvectors don‘t span the whole R3. Why not?