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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
>
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