| [ |
| { |
| "text": "Don't change." |
| }, |
| { |
| "text": "That's the key point about singular values." |
| }, |
| { |
| "text": "If I multiply, so A has an SVD U sigma V transpose." |
| }, |
| { |
| "text": "And QA will have the SVD QU sigma V transpose." |
| }, |
| { |
| "text": "So all I've changed when I multiply by Q, all I changed was the first factor, the first orthogonal factor in the SVD." |
| }, |
| { |
| "text": "I didn't change the sigmas." |
| }, |
| { |
| "text": "They're still sitting there." |
| }, |
| { |
| "text": "And of course, I could do also Q on the other side." |
| }, |
| { |
| "text": "Different Q, same Q or a different Q on the other side would show up here." |
| }, |
| { |
| "text": "And would not change the sigmas, and therefore would not change the Frobenius norm." |
| }, |
| { |
| "text": "So these are important properties of this Frobenius norm." |
| }, |
| { |
| "text": "It looks messy to write down in that form, but it's much nicer in these forms and in that form." |
| }, |
| { |
| "text": "So now if I can just, then we saw that it involves traces." |
| }, |
| { |
| "text": "So let me make a few observations about traces." |
| }, |
| { |
| "text": "So I'll just, we want to be able to play with traces, and that's something we really haven't done." |
| }, |
| { |
| "text": "Here's a fact, that the trace of A transpose B is equal to the trace of B transpose A." |
| }, |
| { |
| "text": "Of course, if B is A, it's clear." |
| }, |
| { |
| "text": "And it's equal to the trace of B A transpose." |
| }, |
| { |
| "text": "So you can do little changes in your matrix without changing the trace." |
| }, |
| { |
| "text": "Let's see why one of these is true." |
| } |
| ] |
