# Generalized Householder Transformations

## Abstract

**:**

## 1. From probabilities to expectations

- (i)
- ${\mathbf{\U0001d5e8}}_{\mathbf{x}}$ is Hermitian; that is, ${\mathbf{\U0001d5e8}}_{\mathbf{x}}={\mathbf{\U0001d5e8}}_{\mathbf{x}}^{\u2020}$;
- (ii)
- ${\mathbf{\U0001d5e8}}_{\mathbf{x}}$ is unitary, that is,$$\begin{array}{c}\hfill {\mathbf{\U0001d5e8}}_{\mathbf{x}}{\mathbf{\U0001d5e8}}_{\mathbf{x}}^{\u2020}={\mathbf{\U0001d5e8}}_{\mathbf{x}}^{\u2020}{\mathbf{\U0001d5e8}}_{\mathbf{x}}={\mathbf{\U0001d5e8}}_{\mathbf{x}}{\mathbf{\U0001d5e8}}_{\mathbf{x}}\\ \hfill =\left(\U0001d7d9-2{\left(\langle \mathbf{x}|\mathbf{x}\rangle \right)}^{-1}|\mathbf{x}\rangle \langle \mathbf{x}|\right)\left(\U0001d7d9-2{\left(\langle \mathbf{x}|\mathbf{x}\rangle \right)}^{-1}|\mathbf{x}\rangle \langle \mathbf{x}|\right)\\ \hfill =\U0001d7d9-4{\left(\langle \mathbf{x}|\mathbf{x}\rangle \right)}^{-1}|\mathbf{x}\rangle \langle \mathbf{x}|+4{\left(\langle \mathbf{x}|\mathbf{x}\rangle \right)}^{-1}|\mathbf{x}\rangle \langle \mathbf{x}|=\U0001d7d9;\end{array}$$
- (iii)
- Hence, ${\mathbf{\U0001d5e8}}_{\mathbf{x}}$ is involutory: ${\mathbf{\U0001d5e8}}_{\mathbf{x}}^{-1}={\mathbf{\U0001d5e8}}_{\mathbf{x}}$;
- (iv)
- The eigensystem of ${\mathbf{\U0001d5e8}}_{\mathbf{x}}$ has two eigenvalues $\pm 1$:
- $-1$:
- For the eigenvector $|\mathbf{x}\rangle $ of ${\mathbf{\U0001d5e8}}_{\mathbf{x}}$, with ${\mathbf{\U0001d5e8}}_{\mathbf{x}}|\mathbf{x}\rangle =\left(\U0001d7d9-2{\left(\langle \mathbf{x}|\mathbf{x}\rangle \right)}^{-1}|\mathbf{x}\rangle \langle \mathbf{x}|\right)|\mathbf{x}\rangle =|\mathbf{x}\rangle -2|\mathbf{x}\rangle =-|\mathbf{x}\rangle $, the associated eigenvalue is $-1$;
- $+1$:
- The remaining $n-1$ mutually orthogonal eigenvectors span the $n-1$-dimensional subspace orthogonal to $|\mathbf{x}\rangle $. Every vector in that subspace has eigenvalue $+1$. (For $n>2$, the spectrum is degenerate.)

Stated differently, for all vectors orthogonal to $|\mathbf{x}\rangle $, the Householder transformation ${\mathbf{\U0001d5e8}}_{\mathbf{x}}$ acts as the identity; for $|\mathbf{x}\rangle $, the Householder transformation ${\mathbf{\U0001d5e8}}_{\mathbf{x}}$ acts as a reflection on the one-dimensional subspace spanned by $|\mathbf{x}\rangle $; - (v)
- Since the determinant of a matrix is the product of its eigenvalues, the determinant of a Householder transformation is $-1$;
- (vi)
- If $\mathcal{C}=\left\{\right|{\mathbf{e}}_{1}\rangle ,|{\mathbf{e}}_{2}\rangle ,\dots ,|{\mathbf{e}}_{n}\rangle \}$ is an orthonormal basis formalizing a context, then the succession of the respective Householder transformations renders negative unity, that is,$$\begin{array}{c}\hfill {\mathbf{\U0001d5e8}}_{{\mathbf{e}}_{1}}{\mathbf{\U0001d5e8}}_{{\mathbf{e}}_{2}}\cdots {\mathbf{\U0001d5e8}}_{{\mathbf{e}}_{n}}=\left(\U0001d7d9-2|{\mathbf{e}}_{1}\rangle \langle {\mathbf{e}}_{1}|\right)\left(\U0001d7d9-2|{\mathbf{e}}_{2}\rangle \langle {\mathbf{e}}_{2}|\right)\cdots \left(\U0001d7d9-2|{\mathbf{e}}_{n}\rangle \langle {\mathbf{e}}_{n}|\right)\\ \hfill =\U0001d7d9-2\underset{\U0001d7d9}{\underset{\u23df}{\left(|{\mathbf{e}}_{1}\rangle \langle {\mathbf{e}}_{1}|+|{\mathbf{e}}_{2}\rangle \langle {\mathbf{e}}_{2}|+\cdots +|{\mathbf{e}}_{n}\rangle \langle {\mathbf{e}}_{n}|\right)}}=-\U0001d7d9.\end{array}$$

## 2. Generalized Operator-Valued Arguments for Mixed States

## 3. Generalized Operations

## 4. Applications beyond the Quantum Domain

## 5. Summary

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Depiction of the Householder transformation ${\mathbf{\U0001d5e8}}_{\mathbf{z}}$ with $|\mathbf{z}\rangle ={\left(\begin{array}{c}1,1\end{array}\right)}^{\u22ba}$ acting on a vector $|\mathbf{x}\rangle ={\left(\begin{array}{c}2,1\end{array}\right)}^{\u22ba}$. The resulting “reflected” vector $|\mathbf{y}\rangle ={\mathbf{\U0001d5e8}}_{\mathbf{z}}|\mathbf{x}\rangle $ and the original vector $|\mathbf{x}\rangle $ have the same length or norm. Its component along $|\mathbf{z}\rangle $ is reversed, whereas its component orthogonal to $|\mathbf{z}\rangle $ remains the same.

**Figure 2.**Orthogonality diagram (hypergraph) of a configuration of observables without any two-valued state, used in a parity proof of the Kochen–Specker theorem presented by Cabello, Estebaranz, and García-Alcaine [6]. One (from 9216) underlaid value assignments represents squares as “+1” and circles as “−1”. A quantum realization is, for example, in terms of 18 orthogonal projection operators associated with the one-dimensional subspaces spanned by the vectors from the origin ${(0,0,0,0)}^{\u22ba}$ to $|{a}_{1}\rangle ={\left(\begin{array}{c}0,0,1,-1\end{array}\right)}^{\u22ba}$, $|{a}_{2}\rangle ={\left(\begin{array}{c}1,-1,0,0\end{array}\right)}^{\u22ba}$, $|{a}_{3}\rangle ={\left(\begin{array}{c}1,1,-1,-1\end{array}\right)}^{\u22ba}$, $|{a}_{4}\rangle ={\left(\begin{array}{c}1,1,1,1\end{array}\right)}^{\u22ba}$, $|{a}_{5}\rangle ={\left(\begin{array}{c}1,-1,1,-1\end{array}\right)}^{\u22ba}$, $|{a}_{6}\rangle ={\left(\begin{array}{c}1,0,-1,0\end{array}\right)}^{\u22ba}$, $|{a}_{7}\rangle ={\left(\begin{array}{c}0,1,0,-1\end{array}\right)}^{\u22ba}$, $|{a}_{8}\rangle ={\left(\begin{array}{c}1,0,1,0\end{array}\right)}^{\u22ba}$, $|{a}_{9}\rangle ={\left(\begin{array}{c}1,1,-1,1\end{array}\right)}^{\u22ba}$, $|{a}_{10}\rangle ={\left(\begin{array}{c}-1,1,1,1\end{array}\right)}^{\u22ba}$, $|{a}_{11}\rangle ={\left(\begin{array}{c}1,1,1,-1\end{array}\right)}^{\u22ba}$, $|{a}_{12}\rangle ={\left(\begin{array}{c}1,0,0,1\end{array}\right)}^{\u22ba}$, $|{a}_{13}\rangle ={\left(\begin{array}{c}0,1,-1,0\end{array}\right)}^{\u22ba}$, $|{a}_{14}\rangle ={\left(\begin{array}{c}0,1,1,0\end{array}\right)}^{\u22ba}$, $|{a}_{15}\rangle ={\left(\begin{array}{c}0,0,0,1\end{array}\right)}^{\u22ba}$, $|{a}_{16}\rangle ={\left(\begin{array}{c}1,0,0,0\end{array}\right)}^{\u22ba}$, $|{a}_{17}\rangle ={\left(\begin{array}{c}0,1,0,0\end{array}\right)}^{\u22ba}$, $|{a}_{18}\rangle ={\left(\begin{array}{c}0,0,1,1\end{array}\right)}^{\u22ba}$, respectively.

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Svozil, K.
Generalized Householder Transformations. *Entropy* **2022**, *24*, 429.
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Svozil K.
Generalized Householder Transformations. *Entropy*. 2022; 24(3):429.
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**Chicago/Turabian Style**

Svozil, Karl.
2022. "Generalized Householder Transformations" *Entropy* 24, no. 3: 429.
https://doi.org/10.3390/e24030429