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From: firdaus janoos on 21 Jul 2010 18:57
Hello,

Does anybody know if the log of the determinant of a matrix is concave
in its matrix arguments ?

That is, if 0< a < 1

log Det(a X_1 + (1-a) X_2 ) >= a log det(X_1) + (1-a) log det(X_2)


Thanks,
-fj
From: Chip Eastham on 21 Jul 2010 20:35
On Jul 21, 6:57 pm, firdaus janoos <fjan...(a)gmail.com> wrote:
> Hello,
>
> Does anybody know if the log of the determinant of a matrix is concave
> in its matrix arguments ?
>
> That is, if 0< a < 1
>
> log Det(a X_1 + (1-a) X_2 )  >= a log det(X_1)  + (1-a) log det(X_2)
>
> Thanks,
> -fj

Yes, log of the determinant is concave on
the restricted domain of positive definite
matrices. Generally, of course, it is not.

regards, chip
From: fj on 21 Jul 2010 22:04
On Jul 21, 8:35 pm, Chip Eastham <hardm...(a)gmail.com> wrote:

>
> Yes, log of the determinant is concave on
> the restricted domain of positive definite
> matrices.  Generally, of course, it is not.
>

That's exactly what I'm looking for - the domain of psd matrices.
Could you give me an outline (or a link) of a proof for this ?

Thanks.
From: Chip Eastham on 22 Jul 2010 15:16
On Jul 21, 10:04 pm, fj <firdaus.jan...(a)gmail.com> wrote:
> On Jul 21, 8:35 pm, Chip Eastham <hardm...(a)gmail.com> wrote:
>
>
>
> > Yes, log of the determinant is concave on
> > the restricted domain of positive definite
> > matrices.  Generally, of course, it is not.
>
> That's exactly what I'm looking for - the domain of psd matrices.
> Could you give me an outline (or a link) of a proof for this ?
>
> Thanks.

See the solution in the last exercise here:

http://www.ifor.math.ethz.ch/teaching/lectures/ot_as09/solution5

As you might guess from the formulation you've
given above:

> log Det(a X_1 + (1-a) X_2 ) >= a log det(X_1) + (1-a) log det(X_2)

proving concavity can be reduced to consideration
of log Det applied to a line between any two
positive definite matrices, so in essence this is
a one-dimensional computation. [Note that the
positive definite domain of matrices is convex
and sometimes referred to as a "cone".]

regards, chip
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