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

```

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