Kalman gain — minimizes trace of covariance: Foundations

This page assumes nothing. Before you can read the parent note Kalman gain, you must be able to read every symbol it throws at you: $x$ , $\overset{x}{^}$ , $e$ , $P$ , $E [\cdot]$ , $tr (\cdot)$ , $H$ , $z$ , $v$ , $R$ , $K$ , and the little superscripts $^{-}$ , $^{+}$ , $^{⊤}$ . We build them one at a time, each anchored to a picture.

0. The stage: "state" and "estimate"

Figure s01 (below): a horizontal number line with a green dot marking the true state $x$ on the right and a blue dot marking our estimate $\overset{x}{^}$ to its left; a red arrow runs from the blue dot to the green dot, labelled $e = x - \overset{x}{^}$ — the error we're about to define.

Figure — Kalman gain — minimizes trace of covariance

1. Error $e$ — measuring "how wrong"

2. Expectation $E [\cdot]$ — the "average over all luck" machine

3. The zero-mean assumption — why $E [e] = 0$ matters

So far we only have the estimation error $e$ ; the sensor and its noise arrive later in §7. So for now the zero-mean idea is stated for $e$ alone (we'll repeat it for the sensor noise once that noise is defined).

4. Variance and covariance $P$ — the "blur" of a guess

When the state has several components (position and velocity), one variance per component isn't enough — errors in different components can be linked. We collect all of them into a matrix. (The $^{⊤}$ "transpose" symbol used just below is defined fully in §6; for now read $e e^{⊤}$ as "the column of errors times the same errors laid out as a row," which builds a square table.)

Figure s02 (below): a wide blue ellipse and a small green ellipse, both centred on a white dot marked $\overset{x}{^}$ , drawn over axes "state component 1" and "state component 2". The wide ellipse is labelled "big blur = big $P$ (unsure)"; the tight one "small blur (sure)" — showing that the size of the ellipse is the amount of uncertainty $P$ encodes.

5. The Euclidean norm $∥ \cdot ∥$ and the trace $tr (\cdot)$

Before the trace, one small piece of notation the parent note leans on:

6. The transpose $^{⊤}$ and matrices $H$ — reshaping and mapping

Figure s03 (below): a blue box on the left labelled "state-space" containing $x = (p, v)$ (position and velocity), a yellow arrow labelled " $H$ (convert)" pointing rightward, and a green box on the right labelled "measurement-space" containing " $H x$ = position only" — illustrating that $H$ drops/mixes state components down into what the sensor can actually read.

7. Measurement $z$ , sensor noise $v$ , noise covariance $R$

8. The superscripts $^{-}$ and $^{+}$ — before vs after

9. The gain $K$ — the slider that ties it all together

Prerequisite map

Equipment checklist

Test yourself — you're ready for the parent note only if you can answer each without peeking.

What does a hat, as in

\overset{x}{^}

, always mean?

"Our best estimate / guess of" — never the true value.

Write the estimation error and say what it points to.

e = x - \overset{x}{^}

; the gap/arrow from our guess back to the truth.

What does

E [\cdot]

compute and why do we need it?

The average over infinitely many random repetitions; noise is random so we compare strategies by average behaviour.

Why do we square the error before averaging?

So positive and negative errors don't cancel —

E [e]

can be 0 while the guess is still wrong;

E [e^{2}]

captures true spread.

What zero-mean assumptions does the filter make, and why?

E [e] = 0

and

E [v] = 0

(unbiased); only then are

P = E [e e^{⊤}]

and

R = E [v v^{⊤}]

true covariances rather than raw second moments.

What does

∥ w ∥

mean?

The Euclidean (straight-line) length

w_{1}^{2} + w_{2}^{2} + \dots

; and

∥ w ∥^{2} = w^{⊤} w

What is

P

in words, and what picture goes with it?

The error covariance

P = E [e e^{⊤}]

; an uncertainty ellipse around the guess — its size is our blur.

What two properties does

P

always have, and why do they matter?

Symmetric (

P = P^{⊤}

) and positive semidefinite (

u^{⊤} P u \geq 0

); these are exactly what let

P

be drawn as a sensible uncertainty ellipse.

Show why

tr (P) = E [∥ x - \overset{x}{^} ∥^{2}]

tr (P) = tr (E [e e^{⊤}]) = E [tr (e e^{⊤})] = E [e^{⊤} e] = E [∥ e ∥^{2}]

What does the matrix

H

do?

Maps state-space into measurement-space;

H \overset{x}{^}

is "what we'd expect the sensor to read."

What are

v

and

R

v

is the random sensor noise inside each reading;

R = E [v v^{⊤}]

is its size (the sensor's own uncertainty).

Distinguish the superscripts

^{-}

and

^{+}

^{-}

= prior, before using the measurement;

^{+}

= posterior, after using it.

Under what assumptions is the posterior ellipse never bigger than the prior?

The linear-Gaussian setting with the optimal gain; it can fail for nonlinear models, bad gains, or non-Gaussian noise.

In one sentence, what is the Kalman gain

K

The 0-to-1 slider setting how much of the sensor's surprise we fold into our guess, chosen to minimize the after-blur.