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A nonzero prearray perturbation changes the posterior but keeps its factor product PSD

Example

Use a scalar state and observation with $P^{-} = R = H = 1$ . The exact measurement prearray is $Z = [\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}] .$ For the perturbation calculation take $E = [\begin{array}{cc} 0 & 1 / 10 \\ 0 & 0 \end{array}], \hat{Z} = Z + E .$ This is a mathematical backward-perturbation example; it does not assert that a particular floating-point QR produces this exact $E$ . The perturbed prearray has the measurement form with $L_{R} = L^{-} = 1$ and observation coefficient $\hat{H} = 11 / 10$ , so the block identity applies to it once its innovation is shown positive below.

Facts & Assumptions

Given: The scalar prearray and the stated nonzero matrix perturbation.

[F1]

A factor perturbation $E$ changes the Gram matrix by at most $2 ∥ Z ∥_{F} ∥ E ∥_{F} + ∥ E ∥_{F}^{2}$ . A factor perturbation gives an explicit Gram-matrix error

[F2]

When the innovation block is positive, the lower-right Schur complement of the block QR Gram matrix is represented as $L_{22} L_{22}^{T}$ . The block QR measurement update gives the canonical Kalman update

Proof

Given: The matrices in the Example.

1.1F2givenalgebra

The exact Gram matrix is $Z Z^{T} = [\begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array}] .$ Its innovation variance is $2$ , its cross covariance is $1$ , and its exact posterior covariance is $1 - 1 / 2 = 1 / 2$ .

2.1F1givenstep 1.1algebra

The perturbed array and its Gram matrix are $\hat{Z} = [\begin{array}{cc} 1 & 11 / 10 \\ 0 & 1 \end{array}], \hat{Z} {\hat{Z}}^{T} = [\begin{array}{cc} 221 / 100 & 11 / 10 \\ 11 / 10 & 1 \end{array}] .$ Here $∥ E ∥_{F} = 1 / 10$ , $∥ Z ∥_{F} = \sqrt{3}$ , and the Gram difference is $[\begin{array}{cc} 21 / 100 & 1 / 10 \\ 1 / 10 & 0 \end{array}]$ , whose Frobenius norm is $\sqrt{641} / 100$ ; it is bounded by [F1]. The perturbed innovation remains positive, and its Schur complement is $1 - \frac{(11 / 10)^{2}}{221 / 100} = 1 - \frac{121}{221} = \frac{100}{221} > 0.$ This differs from $1 / 2$ by $- 21 / 442$ .

3.1F2step 2.1algebra∎

By [F2], the perturbed Schur complement is exactly the represented factor product $L_{22} L_{22}^{T}$ , so it is positive semidefinite; here it is the positive scalar $100 / 221$ . The nonzero prearray perturbation changes the posterior covariance, while the factor representation retains positive semidefiniteness.

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