Logical matrix: Difference between revisions

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In fact, ''U'' forms a [[Boolean algebra]] with the operations [[and (logic)|and]] & [[or (logic)|or]] between two matrices applied component-wise. The complement of a logical matrix is obtained by swapping all zeros and ones for their opposite.

Every logical matrix {{nowrap|1=a = ( a _{i j} )}} has an '''transpose''' {{nowrap|1=a^T = ( a _{j i} ).}} Suppose ''a'' is a logical matrix with no columns or rows identically zero. Then the matrix product, using Boolean arithmetic, a^T a contains the ''m'' × ''m'' [[identity matrix]], and the product a a^T contains the ''n'' × ''n'' identity.

As a mathematical structure, the Boolean algebra ''U'' forms a [[lattice (order)|lattice]] ordered by [[inclusion (logic)|inclusion]]; additionally it is a '''multiplicative lattice''' due to matrix multiplication.