Step-by-Step Guide to Diagonalizing a Matrix with Eigenvalues & Eigenvectors 🧮

Let's compute a full example of Diagonalizing a matrix via eigenvectors and eigenvalues. The steps:
1) Compute the eigenvalues. Take the determinant of A-lambda I and set it equal to zero. The solutions are the eigenvalues
2) Compute the eigenvectors. For each eigenvalue solve (A-lambda I)x=0, which is a homogeneous system
3) Form the P and D matrix. The D matrix is just the eigenvalues down the main diagonal. The P matrix is the corresponding linearly independent eigenvectors for each eigenvalue as columns, in the same order as their eigenvalues were put into D.
4) Finally, we can test that A=PDP^-1 is true by multiplying out the right hand side.

How to VISUALIZE the diagonalization process: https://www.youtube.com/watch?v=EJG6gBeVdfw
How to COMPUTE the diagonalization process: https://www.youtube.com/watch?v=nNwZA1X-C3c&t=373s

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