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 github.com/clum/YouTube/tree/main/LinearAlgebra05 All linear algebra videos in a single playlist (   • Linear Algebra  ) #LinearAlgebra #MatrixMath You can support this channel via Patreon at www.patreon.com/christopherwlum. Thank you for your help!

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