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I am studying definiteness of matrices. Like positive, negative and indefinite matrices. I am confused whether this concept of definiteness is only for symmetric matrices or we do it for any matrix ? Further how this idea is related to quadratic form?

pmath
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Most authors restrict the terms "positive definite" etc. to hermitian matrices (symmetric in the real case), but some do not. See e.g. Wikipedia.

Robert Israel
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  • For hermitian matrices we can decide their definiteness using sign of eigenvalues but what about a matrix which is not hermitian ? – pmath May 28 '18 at 07:35
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    Positive definiteness of a non-hermitian matrix $A$ (for those who allow it) is equivalent to positive definiteness of the hermitian matrix $A+A^*$. – Robert Israel May 28 '18 at 07:38
  • How can we prove or disprove this positiveness of A and B doesn't imply that of AB? – pmath May 28 '18 at 07:45
  • @PrakashNainwal: See https://math.stackexchange.com/questions/113842/is-the-product-of-symmetric-positive-semidefinite-matrices-positive-definite – Hans Lundmark May 28 '18 at 08:58