rearrange/solve matrix equation?

Question

I'm developing a program that uses a linear transformation $$T=Hi*Hk$$ to do a prediction. Where $Hi$ and $Hk$ are 3x3 invertible matrices.

The prediction has an error and I can find a 3x3 correction matrix $Hc$ so that $$T2 = Hc*Hi*Hk$$

now my prediction using $T2$ has no error.

What I want is to introduce my correction matrix $Hc$ into $Hk$ so that I have a new $Hk'$ so that $$T2= Hi*Hk'$$

Is there any close-form way of obtaining $Hk'$ in terms of $Hc,Hi,Hk$ ?

Answer 1

Sure.

$$ H_k' = H_i^{-1} H_c H_i H_k. $$ Since matrix inversion (for $3 \times 3$ matrices) can be carried out explicitly (via Cramer's rule, for instance), this is a closed-form way of finding what you need.

I feel obliged to say that I also think it's a rather bad idea to seek a closed-form solution, for the matrix $H_i$, while invertible, might be badly conditioned, so that any Cramer's-rule approximation of its inverse could be quite bad. But you may have other reasons for wanting a possibly-bad answer, or maybe you know something about $H_i$ that you haven't told us.

Answer 2

$$Hc\cdot Hi \cdot Hk= Hi \cdot Hk' → Hk' = (Hi)^{-1} \cdot Hc\cdot Hi \cdot Hk$$