3

I'm trying to understand a formula for Kalman filters called the Covariance Update Equation. It looks like this: $$ \boldsymbol{ P_{n,n} = \left( I - K_{n}H \right) P_{n,n-1} \left( I - K_{n}H \right)^{T} + K_{n}R_{n}K_{n}^{T} } $$

$P_{n,n-1}$ is a 6x6 matrix and so is $\left( I - K_{n}H \right)$

It made me curious in general, why would you multiply a matrix $P_{n,n-1}$ by another matrix and its transpose? The only other time I ran into this was when rotating a quaternion by matrix multiplication and I wasn't sure why I was doing it then either :)

Thank you

confused
  • 205

1 Answers1

2

If you have a quadratic form $Q(x, x) = \sum_{i, j} a_{ij} x_i x_j,$ a linear coordinate change of $x$ (by $x\to A x$ corresponds to the matrix $Q$ changing by $Q \to A^t Q A.$ Since the covariance matrix is representing a quadratic form, it seems like a reasonable thing to do.

Igor Rivin
  • 25,994
  • 1
  • 19
  • 40