Reconstruct an operator from its packet blocks

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Explore the interactive diagram to change the matrix and input coordinates, then switch individual blocks on and off.

Question: Can we reproduce a matrix operation by calculating its action between smaller pieces? You need only multiplication of two by two matrices.

1. Choose the object and its parts

Work over the real numbers. Take the column vector and coordinate projectors

\[x=\begin{pmatrix}2\\3\end{pmatrix},\qquad P_1=\begin{pmatrix}1&0\\0&0\end{pmatrix},\qquad P_2=\begin{pmatrix}0&0\\0&1\end{pmatrix}. \]

The parts are \(P_1x=(2,0)^{\mathsf T}\) and \(P_2x=(0,3)^{\mathsf T}\). They add to \(x\). Each projector is idempotent, their cross products vanish, and their sum is the identity. Example 1 checks each of these identities and also gives an oblique decomposition.

2. Calculate the operation by two routes

Let

\[T=\begin{pmatrix}1&2\\3&4\end{pmatrix},\qquad Tx=\begin{pmatrix}1\cdot2+2\cdot3\\3\cdot2+4\cdot3\end{pmatrix} =\begin{pmatrix}8\\18\end{pmatrix}. \]

The block \(P_jTP_i\) takes input from packet \(i\) and places its output in packet \(j\). Rows label targets and columns label sources.

Target \(j\), source \(i\)	Contribution \(P_jTP_i x\)
1, 1	\((2,0)^{\mathsf T}\)
1, 2	\((6,0)^{\mathsf T}\)
2, 1	\((0,6)^{\mathsf T}\)
2, 2	\((0,12)^{\mathsf T}\)

Adding the four contributions gives \((8,18)^{\mathsf T}\) again. The reason extends to every vector in this space:

\[\sum_{j=1}^2\sum_{i=1}^2P_jTP_i =(P_1+P_2)T(P_1+P_2)=ITI=T. \]

Example 4 writes out the four block matrices and continues to composition of operations.

3. Remove a block as a control

Omit \(P_2TP_2\). The output becomes \((8,6)^{\mathsf T}\), with missing contribution \((0,12)^{\mathsf T}\). An incomplete block calculation cannot claim the reconstruction identity above.

4. Run the calculation

Follow the download or checkout instructions. From the collection root, run:

python -B examples/komorebi_demonstrations.py packets

Expected output:

Input: (2, 3)
Packet parts: ((2, 0), (0, 3))
Block contributions: ((2, 0), (6, 0), (0, 6), (0, 12))
Direct result: (8, 18)
Reconstructed result: (8, 18)
Without the final block: (8, 6)
PASS: packets

The program checks the matrix identities themselves as well as this vector's output. Its calculations use integers exactly.

5. Follow the result and its evidence

Continue to	Route and scope
Paper	Paper I, for packet families and operator reconstruction.
Exact claim	Exact operator reconstruction. This record links the written statement, proof and transcribed Lean declarations.
Written proof	Paper I, proof beginning at line 512.
Lean	Recorded coverage and setup. The claim record names the existing correspondence. This demonstration adds no Lean verification.
Program	Read or download the complete program.
Library use	First packet program and installation.
More hand calculations	Packets, transfer blocks, coordinate changes.

This illustrates a finite algebraic reconstruction. Infinite sums, convergence, conditioning and application-specific models require their stated hypotheses.

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