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I have matrix difference equation (Riccati equation): $$ X(k+1) = Q + F\left(I - X(k)H^{T}\left(HX(k)H^{T} + R\right)^{-1}H\right)X(k)F^{T}. $$ I have to work on the trace of this matrix.

Is there someway to write $t(k+1) = \operatorname{tr} \biggr(X(k+1)\biggr)$ as a function of $t(k) = \operatorname{tr}\biggr(X(k)\biggr)$? I mean, $t(k+1) = f\biggr(t(k)\biggr)$.

Some suggestions?

Addition:

I forgot that all involved matrices are symmetric and semidefinite positive.

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1 Answer 1

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Unless the matrices involved have very special structures, expressing $\mathrm{tr}\left(X(k+1)\right)$ in terms of $\mathrm{tr}\left(X(k)\right)$ is impossible in general. The reason is simple: different $X(k)$s with the same trace can result in different $X(k+1)$s with different traces. It is easy to construct some simple examples. For instance, suppose all matrices other than the $X$s here are identity matrices. Then $X(k+1)$ have different traces when $X(k)=I$ and when $X(k)=\mathrm{diag}(1+\epsilon,1-\epsilon,1,1,\ldots,1)$, where $\epsilon>0$ is small.

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Well, thank you very much, your arguments are really strong! But what if $X(0)$ is diagonal? No way? –  the_candyman Dec 3 '12 at 18:17
    
In the example I gave, the two different $X(k)$s are already diagonal. Put $k=0$, then you have two examples of diagonal $X(0)$ that result in two different $X(1)$s with different traces. –  user1551 Dec 3 '12 at 18:40
    
Ok ok, thank you again! –  the_candyman Dec 3 '12 at 21:18

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