Find condition which leads to matrix rank two?

Questions
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1

Hey There! I have a Matrix as followed:

{{qd + 2 qo + qv, -qo, -qo, -qd, 0, -qv}, 
{-qo, qd + 2 qo + qv, -qo, -qv, -qd, 0}, 
{-qo, -qo, qd + 2 qo + qv,  0, -qv, -qd}, 
{-qd, -qv, 0,  qd + 2 qu + qv, -qu, qu}, 
{0, -qd, -qv, -qu,   qd + 2 qu + qv, -qu}, 
{-qv, 0, -qd, -qu, -qu, qd + 2 qu + qv}}

I´m now searching for the condition or requirement, that the rank of the matrix is 2. Do you know an attemp for this problem or know how to compute it? I´ve attached the document.

Best Regards,
Martin

2

If the determinant is non-zero, it’s rank two.

3

Look at the eigenvalues of this symmetric matrix:

Ch[x_] := 
 CharacteristicPolynomial[{{qd + 2 qo + qv, -qo, -qo, -qd, 
     0, -qv}, {-qo, qd + 2 qo + qv, -qo, -qv, -qd, 0}, {-qo, -qo, 
     qd + 2 qo + qv, 0, -qv, -qd}, {-qd, -qv, 0, 
     qd + 2 qu + qv, -qu, -qu}, {0, -qd, -qv, -qu, 
     qd + 2 qu + qv, -qu}, {-qv, 0, -qd, -qu, -qu, qd + 2 qu + qv}}, 
   x] // FullSimplify

They are given by:

Res = x //. Solve[Ch[x] == 0, x]

Zero is an eigenvalue, and the other ones consist in 2 (qd + qv), and two pairs of other roots (Res[[3]]==Res[[4]] and Res[[5]]==Res[[6]]). In order that your system is exactly of rank 2, you must have qd + qv=0 and either Res[[3]]=0 or Res[[5]]=0.

4

In[1]:= mat = {{qd + 2 qo + qv, -qo, -qo, -qd, 0, -qv}, {-qo,
qd + 2 qo + qv, -qo, -qv, -qd, 0}, {-qo, -qo, qd + 2 qo + qv,
0, -qv, -qd}, {-qd, -qv, 0, qd + 2 qu + qv, -qu,
qu}, {0, -qd, -qv, -qu, qd + 2 qu + qv, -qu}, {-qv,
0, -qd, -qu, -qu, qd + 2 qu + qv}};

In[6]:= Solve[Det[mat] == 0, {qu, qd, qo, qv}]

During evaluation of In[6]:= Solve::svars: Equations may not give solutions for all "solve" variables.

Out[6]= {{qu -> 0}, {qv -> -qd}, {qv -> (-qd qo - qd qu - 3 qo qu)/(
   qd + qo + qu)}, {qd -> -(1/2) I (-3 I qu + Sqrt[3] qu), 
  qo -> 1/2 (qu + I Sqrt[3] qu)}, {qd -> 1/2 I (3 I qu + Sqrt[3] qu), 
  qo -> 1/2 (qu - I Sqrt[3] qu)}}