Union problem for lattice

The Solution to Union Problems

 

— Given two integer lattices  L(B 1 ) and L(B 2 ) .Compute a basis for the smallest lattice containing both L(B 1 ) and L(B 2 ).

— Assume B 1 = [a 1 a 2 … a n1 ] , B2 = [b 1 ,b 2 …b n2 ] .  Make a new matrix B3 = [a 1 a 2 … a n1 b 1 ,b 2 …b n2 ] . Because a 1 a 2 … a n1 b 1 ,b 2 …b n2   may be linearly dependent , by using elementary matrix transformations , we can find a matrix B4 that its vectors are linearly independent and its vectors can generate all the vectors that can be generated by B3’s vectors.

— B3 =     

 

The first ,the second and the forth vectors are linearly independent. These three vectors are the new base for the lattice(L(B 4 )) that contains both L(B 1 ) and L(B 2 ) .

 

— For B 4 , If we take one of the three vectors away ,  then , the L(B 4 ) will not be able to contain L(B 1 ) and L(B 2 ) . For example :  if we take (1 1 -2 7 ) away , we will not able to generate (2 4 4 9) by using (2 1 4 3) and (-1 1 -6 6).

— So , L(B 4 ) is the smallest lattice containing both L(B 1 ) and L(B 2 )