Lesson 15 of 23 · 3:02
Complement and factor logical matrices
Keep symbolic factors, entrywise operations and valuation separate.
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In this lesson
- 0:00 · Can row and column factors reconstruct a matrix?
- 0:26 · Complement each expression
- 0:45 · Positive valuation connects back to CMs
- 1:04 · Build the left-only factor
- 1:20 · Build the right-only factor
- 1:34 · Reconstruct by entrywise AND
- 1:53 · Why the multiplication convention matters
- 2:22 · Your turn: one factor cell
- 2:47 · Answer: not X and Y
Complete transcript
This transcript includes the answers to prediction prompts.
Show transcript
Can row and column factors reconstruct a matrix? · 0:00
Can row and column factors reconstruct a matrix? A logical matrix stores expressions rather than final truth values. Here M sub X Y is the outer product of ket X and bra Y, with an AND in each cell. We will complement those formulas and reconstruct the matrix from separate factors, keeping entrywise operations distinct from row-by-column multiplication.
Complement each expression · 0:26
Entrywise complement negates the whole formula in each cell. In the upper-left cell, first form X AND Y, then negate that entire conjunction. De Morgan's law gives NOT X OR NOT Y. Apply the same whole-cell operation everywhere; negating one input alone is a different transformation.
Positive valuation connects back to CMs · 0:45
Set X and Y to one. The original logical matrix becomes the AND correspondence matrix. Its entrywise complement becomes NAND. This connects the symbolic complement to the numeric bit-flipping operation we already know. Keep the order of the cells unchanged.
Build the left-only factor · 1:04
Make a left factor with X repeated across the first row and not X repeated across the second. It is the outer product of ket X and the row one, one. This factor carries the row dependence, while deliberately carrying no Y dependence.
Build the right-only factor · 1:20
Make a right factor with Y and not Y in both rows. This is the column one, one times bra Y. It carries the column dependence. We now have two arrays of the same shape, ready to combine at matching positions.
Reconstruct by entrywise AND · 1:34
AND the factors entry by entry. Top left gives X and Y; top right gives X and not Y. The bottom row gives the other two conjunctions. We explicitly label this entrywise AND because it is different from Boolean row-by-column matrix multiplication.
Why the multiplication convention matters · 1:53
We explicitly use entrywise AND for this factorization. If we read the juxtaposition as our AND-and-exclusive-or row-by-column product, the top-left entry becomes X and Y exclusive-or X and Y, which is zero. The two identical terms cancel. The surrounding source prose uses conjunction, but the displayed notation alone does not prove a unique multiplication convention.
Your turn: one factor cell · 2:22
Look at the bottom-left cell of the left and right factors. Which formula results from their entrywise AND? Then predict its value when X is zero and Y is one. First form the expression, then evaluate it.
Prediction pause: 10 seconds.
Answer: not X and Y · 2:47
The bottom-left factors are not X and Y, so their conjunction is not X and Y. At X zero and Y one, both factors are true and the entry is one. Next we will use an entire logical matrix between symbolic state vectors.