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

*To*: mathgroup at smc.vnet.net*Subject*: [mg97129] Re: [mg97055] MatrixRank[m, Modulus -> 5] is broken*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Fri, 6 Mar 2009 04:23:26 -0500 (EST)*References*: <200903041211.HAA27126@smc.vnet.net>

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>