LA75 Eigenspaces, Diagonalization, and Examples

Eigenspaces of linear operators and matrices are introduced. Since these are subspaces, and we can use rank-nullity theorem to find their dimension, we can streamline answering questions about diagonalizability of matrices and linear operators. You should first watch the videos on eigenvalues and eigenvectors and on diagonalizability. Subscribe @Shahriari for more undergraduate math videos.
00:00 Introduction
00:20 Definition: Eigenspaces of linear operators
01:37 Definition: Eigenspaces of square Matrices
03:25 Proposition: An eigenspace for scalar lambda is a subspace consisting of 0 and all the eigenvectors of lambda
07:57 Discussion: Eigenvectors of linear operators vs eigenvectors of their matrices
13:58 Corollary: If a linear operator on an n-dimensional space has n distinct eigenvalues then it is diagonalizable
15:34 The Theorem on independence of eigenvectors generalized
17:00 Example: Deciding diagonalizability of a 3x3 matrix using eigenspaces & rank-nullity theorem
23:09 Example: Finding the eigenvectors
25:44 D = P^{-1}AP illustrated via a commutative diagram
28:20 Finding A^{1000} for a diagonalizable matrix
29:26 A second Example of a 3x3 matrix that is not diagonalizable
32:54 Discussion: Why perturbing a non-diagonalizable matrix will make it diagonalizable?

This is a video in a series of lectures on linear algebra. The series is a rigorous treatment meant for students with no prior exposure to linear algebra. In this full undergraduate course in linear algebra, general vector spaces and linear transformations are emphasized.

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA
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[loveofcode.exfitruso]

Beautiful explanation.
Thank you for this.