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I've got a method for visualising non-zero $2 \times 2$ real matrices (modulo non-zero scalar factor) using the fact that:

  • Nonnegative determinant matrices (modulo non-zero scalar factor) are in 1-to-1 correspondence with hyperbolic cycles in the hyperbolic plane, endowed with orientation
  • Nonpositive determinant matrices (modulo non-zero scalar factor) are in 1-to-1 correspondence with "pencils of lines of equal inclination" in the hyperbolic plane

The significance is that this depicts:

  • The eigenvalues of a matrix
  • Its eigenvectors
  • Its condition number
  • Its Jordan normal form
  • What it's orthogonally similar to
  • Whether it is diagonal
  • Or upper triangular
  • Or lower triangular
  • Or symmetric
  • Or orthogonal

The question arises: Are there other methods for visualising general matrices, which convey properties which are significant for linear algebra? I'm especially interested in techniques valid for $3 \times 3$ matrices.

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  • $\begingroup$ You might be interested in arxiv.org/abs/1702.02131 $\endgroup$
    – Andrei Smolensky
    Commented Feb 8, 2023 at 20:44
  • $\begingroup$ @AndreiSmolensky Interestingly, my approach isn't mentioned $\endgroup$
    – wlad
    Commented Feb 8, 2023 at 21:11

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