Similarity Transformation and Diagonalization

In this video we investigate similarity transformations in the context of linear algebra. We show how the similarity transformation can be used to transform a square matrix into another square matrix that shares properties with the original matrix. In particular, the determinant, eigenvalues, trace, and rank of the two matrices are the same (and the eigenvectors of the similar matrix are related to the eigenvectors of the original matrix).

We then investigate a very specific similarity transformation that can be used to diagonalize the original matrix and place the eigenvalues along the diagonals.

Topics and timestamps:
0:00 – Introduction
0:55 – Definition of a Similarity Transformation
2:00 – Property 1: Same Determinant
5:17 – Property 2: Same Eigenvalues
10:50 – Property 3: Similar Eigenvectors
19:26 – Property 4: Same Trace
22:03 – Property 5: Same Rank
30:36 – Diagonalization
44:55 – Example 1: Non-Defective Matrix
54:27 – Example 2: Defective Matrix
58:16 - Conclusions

Lecture notes and code can be downloaded from https://github.com/clum/YouTube/tree/...

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