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This expression is a second order matrix polynomial equation in the variable L Therefore, the mean-squared posterior state estimation error, which can be computed conveniently as T r(P+ ), is also a function of L Using the properties of the T r function, which are reviewed in Section B2, we can reduce the expression for T r(P+ ) as follows: T r(P+ ) = T r P LHP P H L + L HP H + R L = T r[P ] 2T r[LHP ] + T r[L HP H + R L ] Selection of L to minimize T r(P+ ) requires di erentiation of the scalar function T r(P+ ) with respect to L Using eqns (B55 B56) for the derivative of T r with respect to a matrix, results in d T r(P+ ) = 2 HP dL + 2L HP H + R

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Setting this expression equal to the zero vector and solving for L yields the formula for the Kalman gain vector, which will be denoted by K instead of L: 1 (514) Kk = P Hk Hk P Hk + Rk k k where we have used the fact that P is symmetric Since P and R are positive de nite, the matrix Hk P Hk + Rk is also positive de nite, which k ensures that the inverse in the above formula exist Since Hk P Hk + Rk k is also the second derivative of T r(P+ ) with respect to L, the Kalman gain Kk yields a minimum value for the mean-squared cost

When the Kalman gain is used, a simpli ed equation can be derived for the posterior error covariance matrix P+ The derivation uses the fact that Kk Hk P Hk + Rk = P Hk k k (515)

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