Orthogonal Diagonalization

Note that ‘orthogonal matrix’ is somewhat of a misnomer, since rows (and columns) must not only be orthogonal sets of vectors, but also each vector must be a unit vector, i.e. rows (and columns) must be orthonormal sets of vectors. Better names would be ‘orthonormal matrix’ and ‘orthonormal diagonalization’, but unfortunately the less clear terminology is well established.

Selected Homework 10.  For each (symmetric) matrix A below:

1. find a matrix P that diagonalizes A,

2. use P to diagonalize A,

3. find a matrix Q that orthogonally diagonalizes A,
(Note that before applying Gram-Schmidt, check your eigenvectors for orthogonality.
If they are already pairwise orthogonal, then you need only make them unit vectors to find Q.)

4. use Q to orthogonally diagonalize A (make sure to use Q ^-1 = Q ^t ).