Lecture 43: Non-diagonalizable matrices examples; all symmetric matrices are diagonalizable (set-up for Nicholson Section 8.2)

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Description: Started with a T/F question “Every (square) matrix can be diagonalized”. A better counter-example would have been the matrix [2 1;0 2] so that you can see the eigenvalues 2,2 as separate from the off-diagonal 1. It’s the off-diagonal 1 that stops the matrix from being diagonalizable. What went wrong? If there’s even one eigenvalue for which the algebraic multiplicity is larger than the geometric multiplicity, the matrix will not be diagonalizable.

10:50 --- None of the examples of non-diagonalizable matrices are symmetric. Is this a failure of imagination? Are there symmetric matrices that are non-diagonalizable? No. It’s a theorem --- all symmetric matrices are diagonalizable.

12:00 --- If A is upper triangular, lower triangular, or diagonal is it true that the eigenvalues are precisely on the diagonal? Yes. Did a 4x4 example to show why this is true --- you should verify that it’s true in general.  

15:45 --- Diagonalized a symmetric 3x3 matrix.

24:45 --- After all those computations, nothing had to do with A’s being symmetric. Here’s something interesting: the eigenvectors are mutually orthogonal! This is because A is symmetric. Demonstrated that P^T P is a diagonal matrix. Modified the eigenvectors (made them unit vectors) and this made P^T equal to P^{-1}. (We diagonalized the matrix without having to compute P^{-1} using the inverse matrix algorithm.)

34:30 --- Defined what it means for a square matrix to be orthogonally diagonalizable.