Re: MatrixRank[m, Modulus -> 5] is broken

*To*: mathgroup at smc.vnet.net*Subject*: [mg97320] Re: MatrixRank[m, Modulus -> 5] is broken*From*: dh <dh at metrohm.com>*Date*: Wed, 11 Mar 2009 04:20:37 -0500 (EST)*References*: <200903041211.HAA27126@smc.vnet.net> <goqpbj$lqq$1@smc.vnet.net>

Hi Daniel, what you write is o.k. But Ilan has a more interesting fact. Lets call the original and the augmented matrix m1,m2. The rank of m1 drops from 5 to 4 when one uses modulo 5. That this is possible is not hard to see. However, more interesting: in augmenting m1 mod 5 to m2 mod 5 the rank is increased, whereas in R it is NOT increased. Ilan then reasons that if the additional raw is linear dependent on m1 in R it must be linear dependent on m1 mod 5. However, this is false. The reason is that by taking mod 5 we may map a component to zero, thereby loosing a dimension. Here is an example: m1={{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0},{0,0,0,0,5}} Additional raw: v={0,0,0,0,1} MatrixRank[m1] == MatrixRank[m2] == 5 MatrixRank[m1, Modulus -> 5] == 4 and MatrixRank[m2, Modulus -> 5] ==5 Daniel (Huber) Daniel Lichtblau wrote: > ilan wrote: >> I have a strange problem. >> I ask for MatrixRank of matrix over the reals without Modulus and I get some number, assume 5; >> >> next I add a raw into this matrix and ask for the rank and I get 5; >> >> next I do both calculations using Modulus -> 7 and get 4 and 5. >> >> There is a problem!!! >> if a the additional raw was linear depended over the reals, it m-u-s-t be linear depended over Modulus because a linear combination for this extra row using other rows is valid when we apply Modulus -> 7. >> Therefor I suppose to get 4 and 4. >> >> what is the conclusion? > > A linear dependency over the rationals can evaporate over a field of > positive characteristic (by becoming, in effect, 0==0). Below is an example. > > In[75]:= mat1 = {{3, 4}, {4, 3}}; > mat2 = Append[mat1, {1, 1}]; > {MatrixRank[mat1], MatrixRank[mat2], MatrixRank[mat1, Modulus -> 7], > MatrixRank[mat2, Modulus -> 7]} > > Out[77]= {2, 2, 1, 2} > > Daniel Lichtblau > Wolfram Research >

**References**:**MatrixRank[m, Modulus -> 5] is broken***From:*ilan <ilanorv@cs.bgu.ac.il>