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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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