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I was trying to solve an optimization problem involving rank of a matrix and experimenting with a very simple one. A 2x2 matrix which contains only one parameter a11 is {{a11,2},{2,4}} and I want to minimize the rank of this matrix. Obviously, the matrix can have a minimum rank of 1 when a11 equals 1. But when I use the following commands,

answ = NMinimize[{MatrixRank[{{a11, 2}, {2, 4}}]}, {a11}]

It gives the answer

{2., {a11 -> 0}}

which indicates a rank of 2 with a11=0. I do not quite understand why this is the case. Is it the NMinimize function has difficulties handing the MatrixRank computation? Any comment is welcomed.

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    $\begingroup$ The arguments of NMinimize are being evaluated first, and with symbolic a11, MatrixRank[{{a11, 2}, {2, 4}}] immediately evaluates to the constant 2, so you are effectively doing NMinimize[2, a11]. $\endgroup$ – ilian Nov 11 '16 at 0:01
  • $\begingroup$ For your simple example you can do: Solve[Det[{{a11, 2}, {2, 4}}] == 0, a11] $\endgroup$ – bill s Nov 11 '16 at 0:26
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One way to determine matrix rank is to count the number of zero eigenvalues. For the simple case:

Select[Table[
   Solve[Eigenvalues[{{a, 2}, {2, 4}}][[i]] == 0, a], {i, 2}], Length[#] > 0 &]

This checks to find values of a that give zero eigenvalues and selects all those with non-empty solutions.

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