Learn To Speak Math: Work-Group, Friday 4/12/13

Today we covered the homework problems for section 6.6, #3 and #7 b, c, d. We thank you Ilja for attending and whipping us into shape.

For #3, we actually only covered #3 a. so that we could see how to work this problem:

1. you must show that the matrix is normal over the complex field (or self-adjoint over the reals), which implies that there is a Spectral Decomposition.

2. you must then determine the distinct eigenvalues and their corresponding eigen spaces.

3. you must perform G-S on the eigenvectors so that you can get an orthonormal bases of V consisting of the eigenvectors of T. At this point, you have a set of eigenspaces for each eigenvalue such that the eigenspaces are orthogonal and their direct sum is V. How lovely!

4. Finally you must determine the orthogonal projection of V onto each eigenspace found in 3 above. This is accomplished by creating a matrix from the ONB of each eigenspace. Lets call it W. Then the orthogonal projection of V onto that particular eigenspace is found by matrix multiplication of WW*. Repeat this for each eigenspace. (NB - Please notice in these board notes, that we claimed it was the matrix multiplied by it's transpose, but indeed, it must be the adjoint.)

5. To show that you have actually found the proper orthogonal projections, you must add them together to see that they sum up to the Identity matrix. Or show that the spectral decomposition in fact adds up to the original matrix. See spectral theorem on pg. 401.

Whew! That was a lot of work.

#7 is a standard proof (not to say it's easy, but standard.)

6.6 PROB #3 a