Giving the matrix an antisymmetric component alters its character. The matrix M = [[a,b],[-b,c]] remains positive definite as b increases from 0, but when 4 b^2 > (a - c)^2, the eigenvalues become complex, becoming equal at the crossover point. Furthermore, even while the eigenvalues remain real, the eigenvectors cease to be orthogonal. -- Gene On Tuesday, May 5, 2020, 11:42:15 AM PDT, Cris Moore via math-fun <math-fun@mailman.xmission.com> wrote: if you define a matrix M as positive (semi)definite if v.M.v >= 0 for all vectors v, then you can certainly extend this to non-symmetric matrices. But then it’s just the same as saying that M is positive definite of the average of M and its transpose is, which in your example just means setting b to zero. Cris
On May 5, 2020, at 12:34 PM, Eugene Salamin via math-fun <math-fun@mailman.xmission.com> wrote:
On Tuesday, May 5, 2020, 09:58:03 AM PDT, Dan Asimov <dasimov@earthlink.net> wrote:
A positive definite matrix always means a *symmetric* matrix,
—Dan
Not true. The nonsymmetric matrix [[a,b],[-b,c]] is positive definite when a > 0 and c > 0.
-- Gene