Re: [math-fun] Positive eigenvalues imply positive definite?
A positive definite matrix always means a *symmetric* matrix, and the eigenvectors of a symmetric matrix are always perpendicular to each other as long as their eigenvalues are unequal. A nice condition equivalent to positive definiteness is that all the upper-left k x k submatrices have positive determinant. —Dan
Dan Asimov (hi Dan!) writes: A nice condition equivalent to positive definiteness is that all the
upper-left
k x k submatrices have positive determinant.
Why is this true? If it's true, it has the non-obvious consequence that "all upper-left submatrices have positive determinant" is equivalent to "all lower-right submatrices have positive determinant", right? Jim Propp
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
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
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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
participants (4)
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Cris Moore -
Dan Asimov -
Eugene Salamin -
James Propp