1
$\begingroup$

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}}
$\endgroup$
  • 2
    $\begingroup$ I get a MatrixRank of 18. Rows 16 and 18 are zero. $\endgroup$ – Roman Jul 3 at 11:49
  • $\begingroup$ @Roman Interesting, in version 12.0 for macos tells me, the rank were 16. $\endgroup$ – Henrik Schumacher Jul 3 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 at 13:55
4
$\begingroup$
Unitize@Chop@NullSpace[R]

and

Unitize@Chop@NullSpace[Transpose[R]]

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.