# Kalman smoother derivation subsitution

1. Feb 22, 2013

### MikeLowri123

Hi Im working through the derivation of Kalman smoothing and am stuck on probably a simple substitution problem for the conditional variance of the hidden state x_{t}|x{t+1}, conditioned on all previous observations

Firstly the variance is given by:

$Var[x_{t}|x_{t+1},y_{o},..,y_{t}]=P_{t|t}-P_{t|t}A^{T}P^{-1}_{t+1|t}AP_{t|t}$

with A being the system updat equation and P represent the respective covariance in the relevant states

The substitution is of $L{t}=P_{t|t}A^{T}P^{-1}_{t+1|t}$

And the variance is thus supposed to equal.

$P_{t|t}-L_{t}P_{t+1|t}L^T_{t}$

But working through I obtain:

$P_{t|t}-P_{t|t}A^{T}P^{-1}_{t+1|t}P_{t+1|t}P_{t|t}A^{T}P^{-1}_{t+1|t}$

which condenses to:

$P_{t|t}-P_{t|t}A^{T}P_{t|t}A^{T}P^{-1}_{t+1|t}$

first three terms agree and can be cancelled so Im left with:

$P_{t|t}A^{T}P^{-1}_{t+1|t}$

which should equal

$P^{-1}_{t+1|t}AP_{t|t}$

Due to symmetry however

$P_{t|t}A^{T}=AP_{t|t}$

So:

$AP_{t|t}P^{-1}_{t+1|t}=P^{-1}_{t+1|t}AP_{t|t}$

The question is, does this final term hold, Im not sure if this relationship is commutative or not.