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Can row and column factors reconstruct a matrix?

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A logical matrix stores expressions rather than final truth values.

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Here M sub X Y is the outer product of ket X and bra Y,

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with an AND in each cell. We will complement those
formulas and reconstruct the matrix from separate factors,

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keeping entrywise operations distinct from row-by-column multiplication.

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Entrywise complement negates the whole formula in
each cell. In the upper-left cell, first form X AND Y,

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then negate that entire conjunction. De Morgan's law gives NOT X OR NOT Y.

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Apply the same whole-cell operation everywhere;
negating one input alone is a different transformation.

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Set X and Y to one.

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The original logical matrix becomes the AND correspondence matrix.

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Its entrywise complement becomes NAND.

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This connects the symbolic complement to the numeric bit-flipping
operation we already know. Keep the order of the cells unchanged.

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Make a left factor with X repeated across the first row and not X repeated across the second.

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It is the outer product of ket X and the row one, one.

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This factor carries the row dependence, while deliberately carrying no Y dependence.

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Make a right factor with Y and not Y in both rows. This is the column one, one times bra Y.

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It carries the column dependence. We now have two arrays
of the same shape, ready to combine at matching positions.

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AND the factors entry by entry. Top left gives X and Y;

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top right gives X and not Y. The bottom row gives the other two conjunctions.

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We explicitly label this entrywise AND because it is different
from Boolean row-by-column matrix multiplication.

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We explicitly use entrywise AND for this factorization.

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If we read the juxtaposition as our AND-and-exclusive-or row-by-column product,

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the top-left entry becomes X and Y exclusive-or X and Y, which is zero.

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The two identical terms cancel.

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The surrounding source prose uses conjunction,

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but the displayed notation alone does not prove a unique multiplication convention.

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Look at the bottom-left cell of the left and right factors.
Which formula results from their entrywise AND?

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Then predict its value when X is zero and Y is one. First form the expression, then evaluate it.

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The bottom-left factors are not X and Y, so their conjunction is not X and Y.

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At X zero and Y one, both factors are true and the entry is one.

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Next we will use an entire logical matrix between symbolic state vectors.