Min-plus matrix multiplication

Transcription

We're given two matrices over here, matrix E and matrix D. And they ask us, what is ED, which is another way of saying what is the product of matrix E and matrix D? Just so I remember what I'm doing, let me copy and paste this. And then I'm going to get out my little scratch pad. So let me paste that over here. So we have all the information we needed. And so let's try to work this out. So matrix E times matrix D, which is equal to-- matrix E is all of this business. So it is 0, 3, 5, 5, 5, 2 times matrix D, which is all of this. So we're going to multiply it times 3, 3, 4, 4, negative 2, negative 2. Now the first thing that we have to check is whether this is even a valid operation. Now the matrix multiplication is a human-defined operation that just happens-- in fact all operations are-- that happen to have [? new ?] properties. Now the way that us humans have defined matrix multiplication, it only works when we're multiplying our two matrices. So this right over here has two rows and three columns. So it's a 2 by 3 matrix. And this has three rows and two columns, it's 3 by 2. This only works-- we could only multiply this matrix times this matrix, if the number of columns on this matrix is equal to the number of rows on this matrix. And in this situation it is, so I can actually multiply them. If these two numbers weren't equal, if the number of columns here were not equal to the number of rows here, then this would not be a valid operation, at least the way that we have defined matrix multiplication. The other thing you always have to remember is that E times D is not always the same thing as D times E. Order matters when you're multiplying matrices. It doesn't matter if you're multiplying regular numbers, but it matters for matrices. But let's actually work this out. So what we're going to get is actually going to be a 2 by 2 matrix. But I'm going to create some space here because we're going to have to do some computation. So this is going to be equal to-- I'm going to make a huge 2 by 2 matrix here. So the way we get the top left entry, the top left entry is essentially going to be this row times this product. If you view them each as vectors, and you have some familiarity with the dot product, we're essentially going to take the dot product of that and that. And if you have no idea what that is, I'm about to show you. This entry is going to be 0 times 3, plus 3 times 3, plus 5 times 4. So that is the top left entry. And I already see that I'm going to run out of space here, so let me move this over to the right some space so I have some breathing room. Now we can do the top right entry. This was the top left, now we're going to do the top right. So the top right entry is going to be this row times this column. Notice the entry is getting the row from the first matrix and the column from the second one. That's kind of determining its position. So, once again, is going to be 0 times 4, plus 3 times negative 2, plus 5 times negative 2. And we keep going. The bottom left entry is going to be this row, the second row here times the first column here. So it's 5 times 3, plus 5 times 3, plus 2 times 4. And we're almost done. We just need to multiply or take the dot product of this row with this column right over here. So it's going to be 5 times 4, plus 5 times negative 2, plus 2 times negative 2. And so this is going to be equal to, and we could just evaluate this now. Let's see, 0 times 3 is 0. This is 9 plus 20. This is 29. This all simplified to 29. All of this is 0. This is negative 6. And then this is minus 10. So this all simplifies to negative 16. This right over here, it's 15 plus 15, which is 30 plus 8. So this is 38. And then finally, this is 20 minus 10 minus 4. So this is going to be 6. So this is all going to simplify to 6. So this all became 29, negative 16, 38, and 6. Let's check our answer. And we got it right.