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I have a matrix R which is having a dimension of 20 cross 20. When I extracted the rank of this matrix I got 16. I just wanted to which are these four rows are or columns which are dependent on each other?

 {{-634174. + 78.5 a^2, -9.77228*10^-10, 
  494564. - 61.2197 a^2, -168590. + 20.8681 a^2, 0, 0, 0, 0, 0, 0, 0, 
  0, 0, 0, 0, 0, 0.5, 0, 0.5, 
  0.392699}, {-9.77228*10^-10, -1.01468*10^7 + 78.5 a^2, 
  5.13433*10^6 - 39.7225 a^2, 5.04031*10^6 - 38.9937 a^2, 0, 0, 0, 0, 
  0, 0, 0, 0, 0, 0, 0, 0, -0.707107, 
  0, -0.707107, -1.74472*10^-6}, {494564. - 61.2197 a^2, 
  5.13433*10^6 - 39.7225 a^2, -3.14253*10^6 + 68.0354 a^2, -44.6688 + 
   0.000130837 a^2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.0000454151, 
  0, 0.0000454151, -0.355055}, {-168590. + 20.8681 a^2, 
  5.04031*10^6 - 38.9937 a^2, -44.6688 + 
   0.000130837 a^2, -4.45527*10^7 + 87.9732 a^2, 0, 0, 0, 0, 0, 0, 0, 
  0, 0, 0, 0, 0, -4.51929*10^-6, 0, -4.51929*10^-6, 1.11557}, {0, 0, 
  0, 0, -9.8696*10^9 + 39.25 a^2, -3.33067*10^-7, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 
  0, -3.33067*10^-7, -3.94784*10^10 + 39.25 a^2, 0, 0, 0, 0, 0, 0, 0, 
  0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 
  0, -3.28987*10^9 + 117.75 a^2, -1.11022*10^-7, 0, 0, 0, 0, 0, 0, 0, 
  0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 
  0, -1.11022*10^-7, -1.31595*10^10 + 117.75 a^2, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, -246740. + 39.25 a^2, 
  1.249*10^-10, 0, 0, 0, 0, 0, 0, 0, 0, -1., 0}, {0, 0, 0, 0, 0, 0, 0,
   0, 1.249*10^-10, -2.22066*10^6 + 39.25 a^2, 0, 0, 0, 0, 0, 0, 0, 0,
   1., 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6.1685*10^6 + 39.25 a^2, 0,
   0, 0, 0, 0, 0, 0, -1., 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
  0, -83422.3 + 78.5007 a^2, 
  0.577117 - 0.00105269 a^2, -38514.4 + 76.3093 a^2, -93128.4 + 
   17.5699 a^2, 0, 0, 0, 0, -0.00043019}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 
  0, 0, 0.577117 - 0.00105269 a^2, -633920. + 78.5 a^2, 
  7825.65 - 15.5051 a^2, -392049. + 73.9653 a^2, 0, 0, 0, 0, 
  0.000195972}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
  0, -38514.4 + 76.3093 a^2, 
  7825.65 - 15.5051 a^2, -39620.2 + 78.5 a^2, 
  0.42486 - 0.000518496 a^2, 0, 0, 0, 0, -5.71007}, {0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, -93128.4 + 17.5699 a^2, -392049. + 73.9653 a^2, 
  0.42486 - 0.000518496 a^2, -416082. + 78.5002 a^2, 0, 0, 0, 0, 
  9.98431}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
  0}, {0.5, -0.707107, 0.0000454151, -4.51929*10^-6, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
   0, 0, 0, 0, 0, 0, 0, 0}, {0.5, -0.707107, 
  0.0000454151, -4.51929*10^-6, 0, 0, 0, 0, -1., 1., -1., 0, 0, 0, 0, 
  0, 0, 0, 0, 0}, {0.392699, -1.74472*10^-6, -0.355055, 1.11557, 0, 0,
   0, 0, 0, 0, 0, -0.00043019, 0.000195972, -5.71007, 9.98431, 0, 0, 
  0, 0, 0}}
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    $\begingroup$ I get a MatrixRank of 18. Rows 16 and 18 are zero. $\endgroup$
    – Roman
    Jul 3, 2019 at 11:49
  • $\begingroup$ @Roman Interesting, in version 12.0 for macos tells me, the rank were 16. $\endgroup$
    – Henrik Schumacher
    Jul 3, 2019 at 11:52
  • $\begingroup$ @Roman Oh, I just realized that I had a numerical definition for a hanging around in my Mathematica session. For symbolical a, I also obtain rank a. The rank may differ because MatrixRank for numerical matrices may replace small singular values by 0.. $\endgroup$
    – Henrik Schumacher
    Jul 3, 2019 at 13:55

1 Answer 1

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$\begingroup$ 
Unitize@Chop@NullSpace[R]

and

Unitize@Chop@NullSpace[Transpose[R]]

may give you some information about linear dependency of columns and rows, respectively.

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