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I found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues.

Does this hold for non-symmetric matrices as well?

  • 17
    Caution: there is no general agreement on what "positive definite" means for non-Hermitian matrices. Which definition are you using?2011-11-17
  • 10
    How about: $$[x\ y]\left[\matrix{1&1\cr -1&1}\right]\left[\matrix{x \cr y}\right]=[x\ y]\left[\matrix {x+y\cr-x+y}\right]=(x^2+xy)+(-xy+y^2)=x^2+y^2,$$ and $$ \left|\matrix {1-\lambda &1\cr -1&1-\lambda } \right| =(1-\lambda)^2+1\ne 0. $$2011-11-17
  • 25
    I've said it before and I'll say it again: positive-definite should not be a term that applies to matrices. It should only apply to _quadratic forms_, which are naturally described by _symmetric_ matrices only.2011-11-17
  • 5
    It's a nice sentiment, but the genie's out of the bottle.2013-05-18
  • 2
    There is a nice explanation about non-hermitian positive definite matrices. Please have a look into http://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol20_pp621-639.pdf2015-03-22

2 Answers 2