| [ |
| { |
| "text": "So D is a 3 by 3 matrix with these distances squared." |
| }, |
| { |
| "text": "And it was convenient to use distances squared, because that's what comes into the next steps." |
| }, |
| { |
| "text": "So of course, the distance from each x to itself is 0." |
| }, |
| { |
| "text": "The distance from x distance squared was that." |
| }, |
| { |
| "text": "This one was that, but this one is 6." |
| }, |
| { |
| "text": "So that's the distance matrix." |
| }, |
| { |
| "text": "And we would like to find, the job was to find, and I'm just going to write down, we cannot find x1, x2, and x3 to match those distances." |
| }, |
| { |
| "text": "So what goes wrong?" |
| }, |
| { |
| "text": "Well, there's only one thing that could go wrong." |
| }, |
| { |
| "text": "When you connect this distance matrix D to the matrix x transpose x, you remember the position matrix, maybe I called it G. This is giving, so Gij is the dot product of xi with xj." |
| }, |
| { |
| "text": "Oops, let's make that into a j." |
| }, |
| { |
| "text": "Thank you." |
| }, |
| { |
| "text": "So Gij is the matrix of dot products." |
| }, |
| { |
| "text": "And the great thing was that we can discover what that matrix, that matrix G comes directly from D, comes directly from D. And of course, what do we know about this matrix of cross products?" |
| }, |
| { |
| "text": "We know that it is positive semidefinite." |
| }, |
| { |
| "text": "So what goes wrong?" |
| }, |
| { |
| "text": "Well, just in a word, when we write out that equation and discover what G is, if the triangle inequality fails, we learn that G doesn't come out positive definite." |
| }, |
| { |
| "text": "That's really all I want to say." |
| }, |
| { |
| "text": "And I could push through the example." |
| }, |
| { |
| "text": "G will not come out positive definite if D, if that's D, because it can't." |
| } |
| ] |
