Proof of trace of product of vector and matrices

**gilberto** · May 28th 2012, 07:04 PM

Hi, how can one prove the following identity?

$\mathrm{v^T} M^{-1} D M^{-1}\mathrm{v} = \mathrm{trace }\left( (M^{-1}\mathrm{v}) (M^{-1}\mathrm{v}) ^T D\right)$ ,

where $\mathrm{v}$ is a vector, $M$ is a positive definite and symmetric matrix, and $D$ is a symmetric matrix, and dimension is $n$ .

Thank you for your help.

**girdav** · May 29th 2012, 03:13 AM

Use the fact that $v^TM^{-1}=(M^{-1}v)^T$ (because $M^{-1}$ is symmetric) and that $\operatorname{trace}(ABC)=\operatorname{trace}(CAB )$ .

**gilberto** · June 14th 2012, 04:04 PM

Thank you, I got the identity now. We have to use also, in addition to the facts you mentioned, that the trace of a scalar equals the scalar, right? Thanks!
(Now that the question is solved, how can I mark the post as such? Thanks again)

**girdav** · June 15th 2012, 12:17 AM

That's right.

(you can edit the title in the first post, and add "[Solved]" in)

Thread: Proof of trace of product of vector and matrices

LinkBack

Thread Tools

Display

Proof of trace of product of vector and matrices

Re: Proof of trace of product of vector and matrices

Re: Proof of trace of product of vector and matrices

Re: Proof of trace of product of vector and matrices

Similar Math Help Forum Discussions

vector proof using cross product

Basis for matrices of trace 0

[SOLVED] Linear Algebra, trace similar matrices

Trace of a tensor product

Matrices- Trace

Search Tags