karl
Posts:
397
Registered:
8/11/06


Re: Does this matrix function have real eigenvalues?
Posted:
Jul 28, 2012 2:42 AM


Am 28.07.2012 07:25, schrieb Paul: > On Jul 27, 11:24 pm, karl <oud...@nononet.com> wrote: >> Am 28.07.2012 04:21, schrieb Paul: >> >>> Let G = I  inv(W) * B where W and B are real symmetric positive >>> definite matrices. >>> In simulations it appears that G always has real eigenvalues, though G >>> is not necessarily symmetric or positive definite. >> >>> I wonder if it can be proved in general that G has real eigenvalues? >>> Bonus question: Is there a simple relationship between the eigenvalues >>> of G and those of W and B? >> >>> Best, >>> Paul >> >> G= inv(W)*(WB)= inv(W)*Winv(W)*B=Iinv(W)*B. >> Inv(W) and WB are symmetric, therefore their product will be symmetric too AFAIS. >> >> Karl > > Thanks. Your reexpression of G is correct. However, the product of > symmetric matrices is not necessarily symmetric. >
Ok, too fast. Look here:
http://www.physicsforums.com/showthread.php?t=6676
This condition is fulfilled, since two positive definit symmetric matrices can be jointly diagonalized:
http://en.wikipedia.org/wiki/Positivedefinite_matrix#Simultaneous_diagonalization
You also need: If two symmetric matrices commute, their product is symmetric.
Karl

