3
$\begingroup$

I am stuck on finding $t$ such that:

$\frac{\partial}{\partial t}\|\log_m(M\Lambda^tM^T)\|_F=0$,

where $M$ is $n\times n$ positive definite matrix (not symmetric, not unitary), $\Lambda$ is $n\times n$ diagonal matrix and positive definite, and $t \in (0,1)$.

Found the following link that deals with something similar since $\|A\|_F^2=Tr(AA^T)$ Derivative of matrix involving trace and log . Also this other may help Derivative of a trace w.r.t matrix within log of matrix sums .

Any help is really appreciated.

Thanks!

$\endgroup$
  • $\begingroup$ Solving for what? $\endgroup$ – Bobson Dugnutt Mar 13 '16 at 20:08
  • $\begingroup$ Thanks for the comment. Solving for $t$, sorry. $\endgroup$ – roncreager Mar 13 '16 at 21:02
1
$\begingroup$

That follows is a method using the integral formula for the derivative of the $\log$ function: let $f:A\rightarrow \log(A)$ where $A$ has no eigenvalues in $(-\infty,0]$; then (*) $Df_A:H\rightarrow \int_0^1(t(A-I)+I)^{-1}H(t(A-I)+I)^{-1}dt$.

Let $\Lambda=diag(\lambda_i),R=M\Lambda^tM^T,Z=\log(R),\phi:t\rightarrow tr(ZZ^T)=tr(Z^2)$. Clearly $R'(t)=Mdiag(\log(\lambda_i)\lambda_i^t)M^T$.

$1/2\phi'(t)=tr(ZZ')=tr(Z\int_0^1(u(R-I)+I)^{-1}R'(u(R-I)+I)^{-1}du)=\int_0^1tr(Z(u(R-I)+I)^{-1}R'(u(R-I)+I)^{-1})du$.

Remarks. Clearly $\phi,\phi'$ cannot be formally calculated. For a given $t$, the calculation of $\phi(t)$ is relatively fast. The calculation of $\phi'(t)$ (for a given $t$) is not so easy because we are in front of a complicated rational fraction in $u$; then we need some software as Maple to do the job. In particular, Maple directly gives to you the $\log$ of a matrix. Anyway, the calculation of $\phi'(t)$ is slow. Practically, we can proceed as follows:

  1. Draw an approximtion of the curve $x=\phi(t)$ or print a list of values of $\phi(t)$; locate, in $[a,b]$, a possible local extremum.

  2. Use the regula falsi method for solving $\phi'(t)=0$ in $[a,b]$.

That follows is a toy-example: $M=\begin{pmatrix} -32&27& 99\\-74&8&29\\ -4, &69& 44\end{pmatrix},\Lambda=diag(4,16,7)$.

  1. A local minimum is reached in $[a,b]=[-3.847,-3.845]$.

    1. With one iteration, we obtain $t_0=-3.846234045,1/2\phi'(t_0)\approx -7.10^{-9}$.

With Maple and this toy-example, each calculation of $\phi'(t)$ requires 0"9, with $15$ significant digits.

EDIT. You can see a proof of the above formula (*) in my answer to Calculating the differential of the inverse of matrix exp?

$\endgroup$
  • $\begingroup$ Hi loup blanc, would you mind providing a reference for the integral formula for the derivative of the matrix $\log$ function? $\endgroup$ – roncreager Apr 19 '16 at 11:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.