# Why Bohr was wrong in his response to EPR

Aurélien Drezet(1)

We assess the analysis made by Bohr in 1935 of the Einstein Podolsky Rosen
paradox/theorem. We explicitly describe Bohr’s gedanken experiment involving a
double-slit moving diaphragm interacting with two independent particles and
show that the analysis provided by Bohr was flawed. We propose a different
protocol correcting Bohr’s version that confirms EPR dilemma: Quantum
mechanics is either incomplete or non-local.

##  1 Préambule [hommage à Georges Lochak]

Georges Lochak était un physicien passionné et un écrivain remarquable
vulgarisateur magnifique de la physique. Je pense avoir lu tous ses livres
mais au moins trois ont marqué ma mémoire: il s’agit i) de _Quanta, Grains et
Champs_ [21] coécrit avec Andrade e Silva qui se focalise sur la pensée de de
Broglie et la double solution, ii) sa remarquable biographie de Louis de
Broglie intitulée simplement _Louis de Broglie-Un prince de la science_ [17],
livre merveilleux que je conseille à toute personne curieuse voulant
comprendre l’oeuvre de Broglie (un physicien avec une âme d’historien, un
chercheur du XIX${}^{\textrm{\\`{e}me}}$ siècle exilé au
XX${}^{\textrm{\\`{e}me}}$ 111et cette remarque n’a rien de négative), et
finalement iii) _La géométrisation de la physique_ [18] qui parle de la beauté
des lois et symmétries en physique. Cependant, Lochak fut avant tout le
dernier grand collaborateur de de Broglie (avec Andrade e Silva) et il
défendait sa (libre) pensée dès qu’il le pouvait contre les critiques qui lui
reprochait son retour aux variables cachées (au grand dam des physiciens
nombreux en France qui considéraient et considèrent encore cela comme
rétrograde voire même comme une hérésie). Je voudrais cependant mentionner un
livre qu’il n’a pas écrit mais coédité (avec Simon Diner et Daniel Fargue).
Intitulé _Les incertitudes d’Heisenberg et l’interprétation probabiliste de la
mécanique ondulatoire_ [12] il s’agit en fait d’un cours inédit de Louis de
Broglie réalisé en 1950-1951. Comme Lochak l’explique dans la préface, de
Broglie ne l’a jamais publié (contrairement à son habitude) car:

> Il l’a désavoué, en effet, peu après l’avoir écrit, car il y exposait, pour
> la dernière fois sans critique et d’une manière particulièrement brillante
> et convaincante, l’interprétation de la mécanique ondulatoire selon les
> idées de l’école de Copenhague auxquelles il adhérait encore à l’époque mais
> dont il commença, justement, de douter en relisant son propre texte.[12]

Le livre contient entre autre des commentaires de de Broglie et Lochak suite
au revirement progréssif de la pensée de de Broglie; revirement qui se
cristalisa lorsque celui ci reçu en 1951 un manuscrit de David Bohm [10] où
l’auteur reprenait en l’étendant un peu la vieille théorie de l’onde pilote de
1927 [11] (qui devint par la suite la mécanique ‘Bohmienne’).
Pour le présent article ce qui fondamental c’est que cet ouvrage contient une
description magistrale de la théorie de la mesure quantique de l’époque et
tout particulièrement une analyse des expériences de pensée proposées par
Heisenberg, Einstein et Bohr dans les années 1926-1935. Le célèbre débat
Einstein-Bohr qui fit suite aux 5-6${}^{\textrm{\\`{e}me}}$ congrès Solvay,
qui eurent lieu en 1927 et 1930, est analysé techniquement en détail par Louis
de Broglie. Ce texte est donc précieux pour celles et ceux qui ne
connaitraient que la perspective de Bohr souvent semi-quantitative et très
difficile à suivre. Cependant ce qui nous interessera tout particulièrement
ici c’est la discussion précise du fameux paradoxe d’Einstein Podolsky et
Rosen (EPR) publié en 1935 [13]. Ce paradoxe fut analysé et critiqué par Niels
Bohr dès 1935 dans le même journal [6] mais sa critique est reputée difficile
à lire. Tellement en fait que John Bell lui même affirma qu’il ne comprenait
pas ce que Bohr veut dire dans sa réponse à EPR ([3], chap. 16). Cela motive
le présent travail qui portera sur une analyse plus poussée qu’a l’usuelle du
fameux paradoxe EPR, ou plutôt théorème EPR car c’est bien d’un théorème dont
il s’agit. C’est la réponse de Bohr qui est ici considérée et critiquée en
s’inspirant de l’analyse faite par de Broglie d’une expérience de pensée
proposée par Bohr en 1935 et qui était censée corriger une erreur
d’appreciation et d’analyse des auteurs EPR. Ce travail complète un peu je
pense l’admirable analyse historique faite par de Broglie mais aussi Mara
Beller et Arthur Fine [4]. Pour des raisons d’universalité du propos, et en
trahissant un peu ici le souhait de Lochak de préserver l’usage de la langue
française, le reste du présent article sera rédigé en anglais.

##  2 EPR: ‘Einstein attacks quantum theory’ (New York Times, May 4 1935)

It is (un)famously said that Bohr was victorious in the Einstein-Bohr battle;
the EPR debate being the culminating point of the fight revealing the
grandiose victory of the Copenhagen prophet 222Einstein famously, in private
letters, ironically named Bohr a ‘talmudic’ or ‘mystic’ philosopher and
accused him to behave as a ‘prophet’ preaching complementarity (identified to
a ‘tranquilizing-pill’). De Broglie later assimilated Bohr to a ‘Rembrandt of
physics’ because of his use of clair-obscur.  and the last defeat of Einstein
‘naive’ realism. Nothing can be further away of this unfortunate claim
repeated again and again in the literature and popular press (especially with
the 2022 Nobel prize of physics attributed to Clauser, Aspect and Zeilinger).
To understand this curious issue it is first useful to briefly remind the
content of the debate.
The EPR paradox/theorem concerns nonlocality and completeness in quantum
mechanics. The goal of the EPR article [13] was to show that if we assume the
principle of ‘Einstein locality’ or EPR-locality (that can precisely defined)
quantum mechanics (supposed valid) must be incomplete. More precisely EPR
shows that either quantum mechanics is incomplete or quantum mechanics is
nonlocal i.e., it violates Einstein’s locality principle. Moreover, Bohm in
his 1952 work [10] showed that his own hidden-variables theory able to
complete quantum mechanics and based on the pilot-wave theory of de Broglie is
explicitly non-local. While this doesn’t contradict the EPR results seen as a
theorem Bohm approach was certainly the option ‘which Einstein would have
liked least’ ([3], chap. 1). Of course this was just the beginning of the
story: In 1964 John Bell, based on EPR work, discovered his famous theorem
([3], chap. 2) firmly establishing that quantum mechanics (irrespectively of
being complete or incomplete) must be nonlocal.
The original EPR article focused on the position and momentum observables for
two entangled particles but it is common to use instead the example proposed
by Bohm in 1951 of two spin-$\frac{1}{2}$ particles 1 and 2 entangled in the
singlet state [9]. This is not the strategy that we will follow here and we
will stick to the original EPR configuration. We remind that the original EPR
paper used the two-particle wave function

$\displaystyle\Psi(x_{1},x_{2})=\delta(x_{1}-x_{2}-d)$ $\displaystyle=\iint
dk_{1}dk_{2}\delta(k_{1}+k_{2})\frac{e^{i(k_{1}+k_{2}x_{2}+k_{2}d)}}{2\pi}$
(1)

in the configuration and momentum spaces showing that we have the strong
correlation $x_{1}=x_{2}+d$ and $k_{1}=-k_{2}$. EPR argued (correctly) that if
we measure the spatial coordinate $x_{1}$ of the first particle we know
counterfactually the spatial coordinate $x_{2}$ of the second particle. All
the same we could measure the momentum $k_{2}$ of the second particle and
deduce counterfactually the momentum $k_{1}$ of the first particle. As a
consequence explained EPR we have now precise knowledge of complementary
observables $x_{1},x_{2},k_{1}$ and $k_{2}$ but this is forbidden in the usual
complete quantum mechanical interpretation. Therefore, assuming locality and
the validity of quantum mechanics EPR deduced quantum mechanics must be
incomplete.
It is central to understand that for EPR the countefactual deduction is
developed from Einstein’s locality assumption and not from realism which is
not imposed. More precisely, EPR starts with the Einstein-locality assumption
based on natural features of the classical realist World picture concerning
correlations and relativistic causality. The main idea is that a local
operation made by Alice on particle 1 at space-time point $x_{1}$ should not
influence what is happening to the second particle recorded by Bob at space-
time point $x_{2}$ if the two events are space-like separated (so that no-
signal could propagate between the two points). As Einstein wrote in 1949 :

> But on one supposition we should, in my opinion, absolutely hold fast: the
> real factual situation of the system $S_{2}$ is independent of what is done
> with the system $S_{1}$ which is spatially separated from the former.[14],
> p. 85

The crux of EPR is thus to assume that locality or local-causality is such a
natural assumption that we can use it even independently of realistic
presuppositions (i.e., without involving hidden variables completing quantum
mechanics). This justifies the famous introduction of counterfactual ‘elements
of reality’ by EPR:

> If, without in any way disturbing a system, we can predict with certainty
> (i.e., with probability equal to unity) the value of a physical quantity,
> then there exists an element of physical reality corresponding to this
> physical quantity.[13]

Of course EPR understood perfectly well that a counterfactual reasoning is in
general forbidden in the usual approach to quantum mechanics for a single
particle. This is because for a single particle one could invoke Heisenberg’s
principle to impose a strong form of complementarity and contextuality: It is
impossible to record in one single experiment the position $x$ and momentum
$k$ because the operators $\hat{x}$ and $\hat{k}$ don’t commute. Therefore,
one must choose between one experiment or the other and if one is doing
sequential experiments (like measuring $x$, then $k$ and finally again $x$) it
is known that dispersion will occur in agreement with Heisenberg’s principle.
However, with Einstein-locality EPR found a clean way to somehow circumvent
Heisenberg’s principle limitations. Assuming locality we can know
counterfactually the complementary variables of the two particles even though
we only only recorded $x_{1}$ and $k_{2}$. Therefore assuming locality we get
more information that is usually accepted. In other words, assuming that
quantum is local (QM-L) and complete (QM-C) we deduce quantum mechanics must
be incomplete (QM-IC)! This is a wonderful logical contradiction that can be
formally written:

$\displaystyle\textrm{QM-L}\Rightarrow\textrm{QM-IC},$ i.e.,
$\displaystyle\textrm{QM-C}\Rightarrow\textrm{QM-NL}.$ (2)

where QM-NL (quantum mechanics nonlocal) is the negation of QM-L (i.e.,
$\lnot$ QM-L). Importantly, EPR leads to three physical alternatives:

$\displaystyle(i)\textrm{QM-L}$ $\displaystyle\&$ QM-IC
$\displaystyle(ii)\textrm{QM-NL}$ $\displaystyle\&$ QM-IC
$\displaystyle(iii)\textrm{QM-NL}$ $\displaystyle\&$ QM-C (3)

where (i) was favored by Einstein, (ii) by Bohm (with the pilot wave
interpretation), and (iii) correspond to the orthodox interpretation of
quantum mechanics assumed by Bohr and followers (even if we will see Bohr
didn’t clearly get the point). Also it is important to note that in EPR
incompleteness was obtained through the logical deduction of determinism
meaning that we actually have: QM-L $\Rightarrow$ QM-D, i.e., quantum
mechanics is deterministic 333determinism results from the perfect
correlations $x_{1}=x_{2}+d$ and $k_{1}=-k_{2}$ (assuming locality), and from
that QM-D $\Rightarrow$ QM-IC. The beauty and logic of the EPR
deduction/theorem is often underappreciated and the fact that
counterfactuality and determinism are actually derived and not presupposed by
EPR are still nowadays misunderstood.

##  3 Bohr’s response

The detailed answer of Bohr was rather cryptic and lengthy (Einstein in a
letter to Schrödinger compared Bohr to a talmudic philosopher). Forgetting the
cryptic part, and the long reminiscence concerning the previous battles he won
against Einstein, we see that actually Bohr presented an explicit double-slit
example. This involves two initially independent particles impinging with
normal incidence on a rigid but translatable diaphragm with two narrow
parallel slits separated by the distance $d$. In his analysis Bohr required
the diaphragm to be treated using quantum mechanics so that we actually have a
3-body quantum system. Bohr didn’t give the mathematical details for his
device to work but this can be reconstructed aposteriori from his article (see
also [1, 4, 12, 16, 22, 23]). He actually assumed that just after the 3-body
interaction the quantum state of the whole entangled system reads:

$\displaystyle|\Psi\rangle=\iiint
dx_{1}dx_{2}da\delta(x_{1}-a)\delta(x_{2}+d-a)\Phi_{1}(x_{1})\Phi_{2}(x_{2})\frac{e^{iK_{0}a}}{\sqrt{2\pi}}|x_{1},x_{2},a\rangle$
(4)

where $a$ is the coordinate of the quantum mechanical diaphragm with initial
momentum $K_{0}$. $\Phi_{i}(x_{i})$ are the transversal wave functions of the
two incident particles: These being irrelevant we now write
$\Phi_{i}(x_{i})=1$ in the following. Bohr supposes that the ‘ _momentum of
this diaphragm is measured accurately before as well as after the passing of
the particles’_. This pre and postselection could in principle be easily done
using a Compton-like interaction between the massive diaphragm and a long-
wavelength photon in order to measure the Doppler effect on the photon
frequency (see the Appendix). Moreover, this implies to consider the final
wave-functions (written in different representations):

$\displaystyle\langle
x_{1},x_{2},K|\Psi\rangle=\frac{\delta(x_{2}+d-x_{1})e^{i(K_{0}-K)x_{1}}}{2\pi}$
(5) $\displaystyle\langle
k_{1},k_{2},K|\Psi\rangle=\frac{\delta(K_{0}-K-k_{1}-k_{2})e^{ik_{2}d}}{2\pi}$
(6)

where $K$ is the postselected diaphragm momentum444We have $|K\rangle=\int
da\frac{e^{iKa}}{\sqrt{2\pi}}|a\rangle$, $|K_{0}\rangle=\int
da\frac{e^{iK_{0}a}}{\sqrt{2\pi}}|a\rangle$ (we use a convention where
$\hbar=c=1$).. From Eqs. 5,6 we again find the perfect EPR correlation
$x_{1}=x_{2}+d$ 555We stress that from Eq. 4 we directly get
$(\hat{x}_{1}-\hat{x}_{2})|\Psi\rangle=d|\Psi\rangle$ i.e., $d$ is an
eigenvalue of $\hat{x}_{1}-\hat{x}_{2}$ implying the perfect correlation
$x_{1}=X=x_{2}+d$. However, $|\Psi\rangle$ is not an eigenstate of
$\hat{k}_{1}+\hat{k}_{2}$.  and $k_{2}=K_{0}-K-k_{1}$ which reduces to Eq. 1
if $K_{0}=K$. These conditions actually mean that if we subsequently measure
the coordinates or momentum of the two particles 1 and 2 we obtain the EPR
perfect correlations.
At that stage everything would be OK with Bohr’s analysis: He provided a
mechanical realization of the EPR state starting from a 3-body entangled
system and considering a postselection on the momentum $K$ of the diaphragm.
However, in a subsequent step he considered that in order to measure the
position of particle 1 it would require

> […] to establish a correlation between its behavior and some instrument
> rigidly fixed to the support which defines the space frame of reference.
> Under the experimental conditions described such a measurement will
> therefore also provide us with the knowledge of the location, otherwise
> completely unknown, of the diaphragm with respect to this space frame when
> the particles passed through the slits.

In other words, by projecting on a position state $|X\rangle$ of the
mechanical diaphragm we would instead get

$\displaystyle\langle
x_{1},x_{2},X|\Psi\rangle=\delta(x_{1}-X)\delta(x_{2}+d-X)\frac{e^{iK_{0}X}}{\sqrt{2\pi}}$
(7) $\displaystyle\langle
k_{1},k_{2},X|\Psi\rangle=\frac{e^{i(K_{0}-k_{1}-k_{2})X}}{(2\pi)^{3/2}}e^{ik_{2}d}$
(8)

where now we have clearly $x_{1}=X=x_{2}+d$. Therefore, measuring $X$ (and not
$K$) leads to the perfect knowledge of $x_{1}$ and $x_{2}$. In turn the price
to pay is that we strongly interfered with the postselection procedure leading
originally to Eqs. 5,6, i.e., to the EPR state (this is not surprising since
the operators $\hat{X}$ and $\hat{K}$ do not commute: $[\hat{X},\hat{K}]=i$).
From Eq. 8 we see that the momentum distribution of particles 1 and 2 is very
broad and homogeneous, i.e., $|\langle k_{1},k_{2},X|\Psi\rangle|^{2}$ is a
constant (compare with Eq. 6). A measure of particle-momentum would not lead
to any correlation! Therefore, the new post-selection on the ‘pointer’
$|X\rangle$ actually precludes the EPR analysis. As Bohr wrote:

> By allowing an essentially uncontrollable momentum to pass from the first
> particle into the mentioned support, however, we have by this procedure cut
> ourselves off from any future possibility of applying the law of
> conservation of momentum to the system consisting of the diaphragm and the
> two particles and therefore have lost our only basis for an unambiguous
> application of the idea of momentum in predictions regarding the behavior of
> the second particle.

Bohr’s proposal, immobilizing the diaphragm, actually implies that a measure
of $x_{1}$ precludes a measure $k_{2}$ without disturbance. But this is
certainly not what must occur with independent and local measurements of
$x_{1}$ and $k_{2}$ advocated by EPR! Therefore, Bohr’s example is actually
not a fair realization of the EPR scenario. Clearly, Bohr wanted to show that
it is not possible to realize the EPR scenario with independent measurements
of $x_{1}$ and $k_{2}$ but he was actually mistaken as we show below.
The rigidity of the diaphragm is a central problem in Bohr analysis. The
transfer of momentum is instantaneously transmitted to all parts of the
diaphragm even if the slits separation $d$ is supposed very large666Note that
based on this analysis we get $\langle
x_{1},k_{2},K|\Psi\rangle=\frac{e^{i(K_{0}-K-k_{2})x_{1}}}{2\pi}e^{ik_{2}d}$,
whereas with the Bohr modified state we would instead obtain $\langle
x_{1},k_{2},X|\Psi\rangle=\delta(x_{1}-X)\frac{e^{i(K_{0}-k_{2})X}}{2\pi}e^{ik_{2}d}$.
Bob recording conditional probabilities $P(k_{2}|K)$ (associated with the
first protocol) and $P(k_{2}|X)$ (associated with the second protocol), i.e.,
averaged on all results of Alice, would still see a quantitative difference
showing the disturbance and invasiveness of Bohr’s procedure.. But for EPR
that was the key point of locality: Avoiding any faster than light
communication and influence between independent measurements done on particles
1 and 2. The device of Bohr cannot properly do the job. The problem is that by
projecting on the diaphragm state $|X\rangle$ (i.e., complementary of the
momentum state $|K\rangle$) Bohr didn’t play the EPR game: He destroyed the
initial EPR state by mixing the postselection and the measurement procedure
and therefore precluded the independence of the measurements for $x_{1}$ and
$k_{2}$. Famously Bohr explained that EPR contains ‘ _an ambiguity as regards
the meaning of the expression without in any way disturbing a system_ ’. He
also wrote that there is ‘ _no question of a mechanical disturbance_ ’ in the
experiment protocol but better ‘ _an influence of the very conditions which
define the possible types of predictions_ ’. All this is not self-consistent
since actually Bohr’s proposal for realizing the EPR experiment with a rigid
diaphragm clearly contradicts the independence and locality/separation
condition required in EPR.

Figure 1: Principle of a gedanken EPR-experiment correcting the proposal of
Bohr [6]. Two independent photons 1 and 2 are impinging on a freely moving
double-slit diaphragm. The momenta $K_{0}$ and $K$ of this massive diaphragm
are measured before and after the interaction with the two beams using
Compton/Doppler effects with additional photons (not shown). The transverse
particle position $x_{1}$ of photon 1 is imaged using an objective microscope
(I-P is the image plane). The transverse momentum $k_{2}$ of the second photon
is recorded in the back focal plane (Fourier plane : F-P) of a microscope
objective (note that in principle only one microscope could be used for both
particles).

Moreover, Bohr could have done a correct analysis based on complementarity
with his device if he would have preserved the postselection by the diaphragm
momentum state $|K\rangle$ leading to the good EPR state. This however
requires to not fix the position of the diaphragm (i.e., to not project or
postselect on the state $|X\rangle$). For example, consider particles 1 and 2
are independent photons transmited by the moving slits: By using (see Fig. 1)
two optical microscope objectives focused on the diaphragm plane Alice and Bob
could decide to image independently the position $x_{1}$ of the particle 1
(i.e., by recording photons 1 in the image plane of the microscope objective)
and the momentum $k_{2}$ of the particle 2 (i.e., by recording photons 2 in
the back-focal plane of the microscope objective). This procedure would not
interfere with the post selection leading to the EPR state, i.e., Eqs. 5, 6.
The EPR diagnostic concerning locality and completeness in quantum mechanics
is rigorously valid in this amended gedanken experiment and the deduction of
the authors cannot said to be faulty contrarily to Bohr’s claims.

##  4 Conclusion

In his later analysis [7, 8] Bohr’s emphasized the indivisibility of quantum
phenomena, i.e., ‘the quantum wholeness’ without invoking the details of his
1935 EPR refutation and it has been said that he somehow modified or corrected
his first view about complementarity based mostly on local disturbance and the
Heisenberg principle (this has been claimed in [4]). It is however not very
clear that he fully appreciated the real content of the EPR paradox in his
writings [16, 4, 15]. Clearly, that was not so in 1935. Interestingly, the
best argumentation that Bohr could have given was already in the EPR paper
where it is written at the end:

> One could object to this conclusion on the grounds that our criterion of
> reality is not sufficiently restrictive. Indeed, one would not arrive at our
> conclusion if one insisted that two or more physical quantities can be
> regarded as simultaneaous elements of reality only when can be
> simultaneously measured or predicted.

This is actually alternative (iii) of Eq. 3 that EPR disliked very much since
‘ _No reasonable definition of reality could be expected to permit this_ ’. It
must be observed that Bohr article was not the first response to EPR appearing
in print in the physical review. Arthur E. Ruark in a very short paper focused
on the previous quotation of EPR and developed the strongly positivist
conclusion (iii):

> This [EPR] conclusion is directly opposed to the view held by many
> theoreticians, that a physical property of a given system has reality only
> when it is actually measured, and that wave mechanics gives a faithful and
> complete description of all that we can learn from measurements.

Bohr wished a better response: He wanted to convince Einstein the realist of
the inevitability of complementarity. Bohr’s positivism was a pragmatic one
and his goal was to explain why experimental/theoretical considerations tied
to the ‘ _finite interaction between object and measuring agencies_ ’ actually
impose his complementarity approach of quantum mechanics that ‘ _fulfill,
within its scope, all rational demand on completeness_ ’. However, the
particular gedanken experiment he provided was flawed and couldn’t be used to
demonstrate a contradiction. It is also interesting to point out that Bohr
didn’t explicitly speak of nonseperability or nonlocality. For Bohr the key
feature was indivisibility or impossibility to analyze the quantum system
without taking into account the whole experimental setup. Bohr refuted the
mere possibility to understand or analyze quantum phenomena within a single
causal and spatio-temporal representation that would contradict the spirit of
complementarity. However, this issue was not in conflict with the EPR result
since alternative (iii) -considered by Einstein as non plausible- was clearly
discussed in [13].
Retrospectively can we say that Einstein won the debate against Bohr? Alas,
this is not so simple because actually the alternative (i) that Einstein
favored was later shown to be untenable by Bell in 1964: Quantum mechanics
must be nonlocal irrespectively of being complete or incomplete. Still, it
must be said that locality or more precisely local-causality is a compound
principle merging locality and causality, parameter setting and outcome
independence. There are several possible loopholes that can be used in order
to built up hidden-variables theories agreeing with the results of Bell’s
theorem and give sense to the spooky nonlocality.

##  5 Appendix

In his book _Les incertitudes d’Heisenberg et l’interprétation probabiliste de
la mécanique ondulatoire_ [12] de Broglie discussed the measurement of the
velocity of an electron using the Doppler effect. The idea is to consider a
photon-electron collision. If $\omega$, $\omega^{\prime}$ (respectively $v$,
$v^{\prime}$ ) denote the photon energy (respectively electron velocity)
before and after the collision we have

$\displaystyle\omega+\frac{1}{2}mv^{2}=\omega^{\prime}+\frac{1}{2}mv^{\prime
2}$ $\displaystyle mv-\omega=mv^{\prime}+\omega^{\prime}.$ (9)

From this we deduce

$\displaystyle\omega-\omega^{\prime}=\frac{1}{2m}[(\omega+\omega^{\prime})^{2}-2mv(\omega+\omega^{\prime})]\simeq
2\frac{\omega^{2}}{m}-2v\omega$ (10)

containing a Compton and Doppler recoil. In the limit where the electron mass
is very large we obtain $\omega-\omega^{\prime}\simeq-2mv\omega$ which is
Doppler formula. The result is left unchanged if we replace the electron by
the diaphragm with very large mass $M$. Therefore, measuring the photon
frequency shift while illuminating the quantum diaphragm we can deduce its
momentum $K_{0}$ and $K^{\prime}$ before and after the interaction with the
two particles 1 and 2.

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# On the Periodicity of Singular Vectors and the Holomorphic Block-Circulant
SVD on the Unit Circumference

Giovanni Barbarino
Mathematics and Operational Research Unit, Faculté polytechnique,
Université de Mons, Belgium<EMAIL_ADDRESS>
###### Abstract

We investigate the singular value decomposition of a rectangular matrix that
is analytic on the complex unit circumference, which occurs, e.g., with the
matrix of transfer functions representing a broadband multiple-input multiple-
output channel. Our analysis is based on the Puiseux series expansion of the
eigenvalue decomposition of analytic para-Hermitian matrices on the complex
unit circumference. We study the case in which the rectangular matrix does not
admit a full analytic singular value factorization, either due to partly
multiplexed systems or to sign ambiguity. We show how to find an SVD
factorization in the ring of Puiseux series where each singular value and the
associated singular vectors present the same period and multiplexing
structure, and we prove that it is always possible to find an analytic pseudo-
circulant factorization, meaning that any arbitrary arrangements of
multiplexed systems can be converted into a parallel form. In particular, one
can show that the sign ambiguity can be overcome by allowing non-real
holomorphic singular values.

##  1 Introduction

Since its introduction [17] the singular value decomposition (SVD) has been
extensively used for a wide range of signal processing problems [5, 10, 20],
from optimal precoder and equaliser design for multiple-input multiple-output
(MIMO) channel matrices of complex gain factors [21] to broadband problems
whose channel matrix is determined by impulse responses [7]. In the last case,
though, the SVD ignores time or frequency correlations in the signals leading
to loss of spectral coherence [25].

The focus has therefore shifted to the computation of the SVD for matrices
with entries in functional rings and fields, in the hope to retain more
information and regularity from the signals without having to process each
time step or frequency separately. In this context, several algorithms for a
polynomial SVD (PSVD) of possibly rectangular matrices have been recently
developed, performing an approximate factorisation into Laurent-polynomial
matrices. The use of PSVD algorithms extends from generic problems [8, 19],
finding application in various practical scenarios such as MIMO communications
[1, 2, 6, 12, 18, 22], the equalisation of filter bank-based multi-carrier
systems [9, 14], or broadband beamforming [27].

Despite the fact that the algorithms in [4, 8, 13, 15, 16] are proven to
converge to a diagonal matrix, it has not been until recently [26] that
questions about existence and uniqueness of such a decomposition have been
raised to understand what these algorithms are actually converging to. The
purpose of this paper is to continue the investigation about the existence of
the analytic SVD of a (possibly rectangular) matrix, and study how changing
the constraints on the wanted factorization may lead to more regular
decompositions. The main contribution, in a signal processing context, is that
any system can be brought by paraunitary operations into a block-diagonal form
containing pseudo-circulant matrices. This is quite profound: it means that
any arbitrary arrangement (concatenations, nesting, etc.) of multiplexed
systems can be converted and decoupled into a parallel form.

Consider a (possibly rectangular) matrix ${\bf A}(z)$ matrix that is
holomorphic on the unit circumference $S^{1}$ . It can be equivalently
represented by the Laurent time series with coefficients ${\bf
A}[n]\laplace{\bf A}(z)$ or by the analytic and $2\pi$ periodic matrix on the
real line ${\bf A}(\Omega):={\bf A}(e^{j\Omega})$. When $L$ is a positive
integer, then with ${\bf B}(z^{1/L})$ we denote a matrix whose entries can be
written as Laurent series in the variable $z^{1/L}$, called Puiseux series
with index $L$, that converges on an annulus around $S^{1}$, and can be
equivalently represented by the analytic and $2\pi L$ periodic matrix on the
real line ${\bf B}(\Omega):={\bf B}(e^{j\Omega/L})$. See [3, Section 2.1] for
a more detailed discussion on the topic. From now on, any matrix, vector or
scalar function depending on some complex variable $z$ or $z^{1/L}$ is
intended to be analytic with respect to the relative variable at least on
$S^{1}$.

In [26] and [3] it has been proven that any rectangular matrix ${\bf A}(z)$
admits an SVD

${\bf A}(z)={\bf U}(z^{1/L}){\bf\Sigma}(z^{1/L}){\bf V}^{P}(z^{1/L})$ (1)

for some positive integer $L$, where ${\bf\Sigma}(z^{1/L})$ is a diagonal
matrix and ${\bf U}(z^{1/L})$, ${\bf V}(z^{1/L})$ are para-unitary matrices,
i.e. ${\bf U}(z){\bf U}^{P}(z)={\bf I}$ with the para-Hermitian operator
$\\{\cdot\\}^{P}$ performing a time reversal and complex conjugation ${\bf
U}^{P}(z):=[{\bf U}(1/\overline{z})]^{H}$. Moreover, the diagonal entries of
${\bf\Sigma}(z^{1/L})$ are real for any $z\in S^{1}$, and thus their absolute
values are exactly the singular values of ${\bf A}(z)$. The difference with a
traditional SVD is that we allow for the singular values to be also negative,
which is a necessary requisite in order to retain the analyticity of the
factorization.

The parameter $L$ comes from the presence of multiplexed singular values and
(an odd number of) zeros of the singular values $\sigma_{i}(z^{1/L})$ in
${\bf\Sigma}(z^{1/L})$. In fact, the singular values can be partitioned into
orbits such that $k$-multiplexed singular values are in orbits of cardinality
$k$, and

$\displaystyle\sigma_{i+1}(\Omega)\equiv\sigma_{i}(\Omega+2\pi),\qquad\forall
i=1,\dots,k-1,$
$\displaystyle|\sigma_{1}(\Omega)|\equiv|\sigma_{k}(\Omega+2\pi)|\equiv|\sigma_{1}(\Omega+2\pi
k)|,$ (2)

where $\sigma_{i}(\Omega):=\sigma_{i}(e^{j\Omega/L})$ and
$|\sigma_{i}(\Omega)|\not\equiv|\sigma_{j}(\Omega)|$ for every $i\neq j$. In
case $\sigma_{1}(\Omega)\equiv\sigma_{k}(\Omega+2\pi)$, we say that the
singular values $\sigma_{i}$ in this orbit are $k$-multiplexed and they are
analytic in $z^{1/k}$. If instead
$\sigma_{1}(\Omega)\equiv-\sigma_{k}(\Omega+2\pi)$ we say that the singular
values are signed $k$-multiplexed and in this case they are analytic in
$z^{1/2k}$.

As a way to get rid of the multiplexed index and return to classical
holomorphic functions, it has been shown in [24] and [3] that any square para-
Hermitian matrix ${\bf A}(z)={\bf A}^{P}(z)$ admits a decomposition

${\bf A}(z)={\bf U}(z){\bf C}(z){\bf U}^{P}(z),$

where ${\bf U}(z)$ is para-unitary and ${\bf C}(z)$ is block diagonal and each
block is pseudo-circulant, i.e. for each block of size $N$ there exist some
functions $\phi_{0}(z),\dots,\phi_{N-1}(z)$ such that the block has the form

$\begin{bmatrix}\phi_{0}(z)&z^{-1}\phi_{N-1}(z)&\dots&z^{-1}\phi_{1}(z)\\\
\phi_{1}(z)&\phi_{0}(z)&\ddots&\vdots\\\
\vdots&\ddots&\ddots&z^{-1}\phi_{N-1}(z)\\\
\phi_{N-1}(z)&\dots&\phi_{1}(z)&\phi_{0}(z)\end{bmatrix}$

and each block coincide with a different multiplexed orbit. With respect to
the above pseudo-circulant eigendecomposition for para-Hermitian matrices,
whose blocks are determined by the multiplexing of the eigenvalues, when
dealing with singular values we have also to be wary of possibly signed
multiplexed orbits, that are less regular than the not signed ones.

Here we show the existence of different "singular values" decompositions for
any (possibly rectangular) matrix ${\bf A}(z)$, where we relax some
constraints on the structure of the SVD in order to gain more regularity.

First of all, we refine the decomposition in [3, 26] and find an analytic SVD

$\textbf{A}(e^{j\Omega})=\textbf{U}(\Omega){\bf\Sigma}(\Omega)\textbf{V}(\Omega)^{H},$

where the diagonal matrix ${\bf\Sigma}(\Omega)$ may have negative entries and
contains the singular values of ${\bf A}(\Omega)$ up to the sign, and for each
nonzero singular value $\sigma(\Omega)$ that is (signed) $k$-multiplexed, the
associated left and right singular vectors sports the same (signed)
multiplexed behaviour.

Then we allow for complex singular values in order to get rid of the signed
multiplexed orbits, i.e. we find an analytic SVD

$\textbf{A}(e^{j\Omega})=\textbf{U}(\Omega){\bf
S}(\Omega)\textbf{V}(\Omega)^{H},$

where ${\bf S}(\Omega)$ is a diagonal and complex matrix, with diagonal
entries $s_{i}(\Omega)$ whose absolute values correspond with the singular
values of $\textbf{A}(e^{j\Omega})$. The functions $s_{i}(\Omega)$ are still
multiplexed, but they are no more signed multiplexed.

Leveraging on the latest decomposition, we then prove that, even in the
rectangular case, any ${\bf A}(z)$ admits a pseudo-circulant decomposition of
the form

${\bf A}(z)={\bf U}(z){\bf C}(z){\bf V}^{P}(z),$

where ${\bf C}(z)$ is block diagonal and each block is pseudo-circulant.

##  2 Periodicity of Singular Vectors

Let ${\bf A}$ be a (possibly rectangular) matrix with distinct non-zero
singular values $\sigma_{1},\dots,\sigma_{p}$, where each singular value has
multiplicity $q_{i}$. In the SVD ${\bf A}={\bf U}{\bf\Sigma}{\bf V}^{H}$, the
matrix ${\bf\Sigma}$ is diagonal, real, nonnegative and contains the singular
values $\sigma_{i}$ with the respective multiplicities on the diagonal, plus
possibly some zeros. The factorization can be equivalently be written in
dyadic notation as

${\bf A}=\sum_{i=1}^{p}\sigma_{i}{\bf U}_{i}{\bf V}_{i}^{H},$ (3)

where ${\bf U}_{i}$ are the $q_{i}$ columns of ${\bf U}$ containing the left
singular vectors relative to $\sigma_{i}$ and ${\bf V}_{i}$ are the $q_{i}$
columns of ${\bf V}$ containing the right singular vectors relative to
$\sigma_{i}$. Even if the SVD is in general not unique, a known result is that
each factor ${\bf A}_{i}:={\bf U}_{i}{\bf V}_{i}^{H}$ in (3) is uniquely
determined by $\sigma_{i}$. Moreover, suppose the singular values in
${\bf\Sigma}$ are sorted so that repeated singular values are consecutive on
its diagonal (not necessarily in some order). If now we take the two EVDs
${\bf A}{\bf A}^{H}={\bf Q}_{1}{\bf D}_{1}{\bf Q}_{1}^{H}$ and ${\bf
A}^{H}{\bf A}={\bf Q}_{2}{\bf D}_{2}{\bf Q}_{2}^{H}$ where ${\bf D}_{1}$,
${\bf D}_{2}$ are real, diagonal, nonnegative and present the same nonzero
eigenvalues $\lambda_{i}=\sigma_{i}^{2}$ on the same positions as the relative
singular values appear in ${\bf\Sigma}$, then

${\bf A}={\bf Q}_{1}{\bf\Sigma}{\bf\Psi}{\bf Q}_{2}^{H},$ (4)

where ${\bf\Psi}$ is a square, block diagonal and unitary matrix, each block
relative to a repeated singular value, with size equal to its multiplicity.

When dealing with analytic matrices ${\bf A}(t)$ depending on on a real
variable $t$, we already know that there exists a ’SVD’ decomposition:

######  Theorem 1 (Analytic SVD on a real interval, [11]).

For an $M\times N$ matrix ${\bf A}(t)$ that is analytic on some interval of
$\mathbb{R}$, a decomposition

${\bf A}(t)={\bf U}(t){\bf\Sigma}(t){\bf V}(t)^{H}$ (5)

exists with analytic unitary ${\bf U}(t)$ and ${\bf V}(t)$, and analytic,
diagonal, real-valued ${\bf\Sigma}(t)$.

The decomposition (5) is not exactly a SVD since the diagonal elements of
${\bf\Sigma}(t)$ are permitted to be negative real numbers, but their absolute
values coincide with the singular values of ${\bf A}(t)$. The negativity of
the singular values, though, is an essential hypothesis to guarantee the
analiticity of the decomposition, so from now on we call (5) the analytic SVD
of ${\bf A}(t)$. The necessity of this hypothesis is shown by the following
example.

######  Example 1.

Consider the $1\times 1$ matrix ${\bf A}(t)=2\cos(t/2)e^{jt/2}$. A unitary
$1\times 1$ matrix is just a unit norm complex number so in (5) we can call
$\gamma(t):={\bf U}(t){\bf V}(t)^{H}$, $|\gamma(t)|=1$,
$\sigma(t):={\bf\Sigma}(t)\in\mathbb{R}$ and find that
$2\cos(t)e^{jt/2}=\gamma(t)\sigma(t)$ from which
$|2\cos(t/2)|=|\sigma(t)|=\pm\sigma(t)$ for every $t$. Since $\gamma(t)$ must
be analytic, the only two possible factorization are $\sigma(t/2)\equiv
2\cos(t),\gamma(t)\equiv e^{jt/2}$ or
$\sigma(t)\equiv-2\cos(t),\gamma(t)\equiv-e^{jt/2}$ and in both cases
$\sigma(t)$ is negative on some real multi-interval.

Consider now the matrix ${\bf A}(z)$ and its $2\pi$ periodic counterpart ${\bf
A}(\Omega)={\bf A}(e^{j\Omega})$. From (5) there exists an analytic SVD ${\bf
A}(\Omega)={\bf U}(\Omega){\bf\Sigma}(\Omega){\bf V}(\Omega)^{H}$, but we do
not have any prior information about the periodicity of the factors (or if
they are periodic at all). From [26] it has been shown that such a
decomposition exists where all the factors are $2\pi L$ periodic for some
positive integer $L$, so that one can reconstruct the decomposition ${\bf
A}(z)={\bf U}(z^{1/L}){\bf\Sigma}(z^{1/L}){\bf V}^{P}(z^{1/L})$ in Puiseux
series of index $L$. In the following we study the periodicity of each
singular value and vector, and we show that when the period of
$\sigma_{i}(\Omega)$ is less than $2\pi L$ then its left and right singular
vectors can present the same period of $\sigma_{i}(\Omega)$ or even half of
it.

###  2.1 Multiplexing and Sign Changes

The necessity of using the Puiseux series to find a SVD as in (1) is due to
the possibility of the singular values being multiplexed or having a sign
change. Recall that the sign problem is already present in the analytic SVD
(5) and the matrix in Example 1 already exhibits this issue:

######  Example 2.

Take ${\bf A}(z)=1+z$, whose associated matrix ${\bf
A}(\Omega)=1+e^{j\Omega}=2\cos(\Omega/2)e^{j\Omega/2}$ is the same as Example
1. Its only possible singular value, up to the sign, is thus
$\sigma(z^{1/2})=2\cos(\Omega/2)=z^{j\Omega/2}+z^{-j\Omega/2}$.

Given a matrix ${\bf A}(z)$ and the analytic SVD ${\bf A}(\Omega)={\bf
U}(\Omega){\bf\Sigma}(\Omega){\bf V}(\Omega)^{H}$, we can always suppose that
all the analytic singular values $\sigma_{i}(\Omega)$ are nonnegative in a
small enough right interval of zero, i.e. $[0,\epsilon]$, up to a constant
$\pm 1$ diagonal matrix that can be multiplied to ${\bf U}(\Omega)$ and
${\bf\Sigma}(\Omega)$. In particular, this choice assures us that if
$\sigma_{i}(\Omega)\not\equiv\sigma_{j}(\Omega)$, then
$|\sigma_{i}(\Omega)|\not\equiv|\sigma_{j}(\Omega)|$, because otherwise they
would coincide on the interval $[0,\epsilon]$. Call then
$\sigma_{1}(\Omega),\dots,\sigma_{p}(\Omega)$ the distinct nonzero singular
values of ${\bf A}(\Omega)$ and, if it exists, its zero singular value
$\sigma_{0}(\Omega)\equiv 0$.

Let now $\Omega_{0}$ be a point such that all distinct singular values
$\sigma_{i}(\Omega)$ have different absolute value when evaluated on
$\Omega_{0}$. Notice that due to the analiticity of the singular values,
$\Omega_{0}$ can be almost every real number, since the set of values $\Omega$
for which two distinct singular values take the same absolute value is
discrete.

Notice that the absolute values of the diagonal entries of
${\bf\Sigma}(\Omega)$ are always the classical singular values of ${\bf
A}(\Omega)$, and since it is $2\pi$ periodic, then ${\bf A}(\Omega_{0})={\bf
A}(\Omega_{0}+2\pi)$ so ${\bf\Sigma}(\Omega_{0})$ and
${\bf\Sigma}(\Omega_{0}+2\pi)$ must contain the same diagonal entries up to
signs and permutations. In particular, this shows that

  * •

the number of zero diagonal entries of ${\bf\Sigma}(\Omega_{0})$ and
${\bf\Sigma}(\Omega_{0}+2\pi)$ is the same and equals the number of zero
singular values of ${\bf A}(\Omega)$,

  * •

if $|\sigma_{i}(\Omega_{0}+2\pi)|\neq|\sigma_{i}(\Omega_{0})|$ then there
exists $j\neq i$ such that
$|\sigma_{j}(\Omega_{0})|=|\sigma_{i}(\Omega_{0}+2\pi)|$ and they have the
same multiplicity $q_{i}=q_{j}$,

  * •

there exists a unique permutation $\tau$ of the indexes of the nonzero
singular values such that
$|\sigma_{\tau(i)}(\Omega_{0})|=|\sigma_{i}(\Omega_{0}+2\pi)|$ for all
$i=1,\dots,p$,

  * •

since $\Omega_{0}$ can be almost every real number, and all
$|\sigma_{i}(\Omega)|$ are at least continuous, then
$|\sigma_{\tau(i)}(\Omega)|\equiv|\sigma_{i}(\Omega+2\pi)|$.

Up to a renaming of the indexes, we can always suppose
$\tau=(1,\dots,k_{1})(k_{1}+1,\dots,k_{1}+k_{2})\dots(p-k_{R}+1\dots,p)$,
meaning that the permutation $\tau$ partitions the $p$ distinct singular
values into $R$ sets, or orbits, indexed by $\nu=1,\dots,R$, each of
cardinality $k_{\nu}$, and cyclically shifts them. Moreover, up to changing
the global sign of some of the singular values, we can drop almost all the
absolute values in the above relation, except the one that links the last
element of each cycle to the first. For example, looking at the first set, we
find that

$\displaystyle\sigma_{i+1}(\Omega)\equiv\sigma_{i}(\Omega+2\pi),\qquad\forall
i=1,\dots,k_{1}-1,$
$\displaystyle|\sigma_{1}(\Omega)|\equiv|\sigma_{k_{1}}(\Omega+2\pi)|\equiv|\sigma_{1}(\Omega+2\pi
k_{1})|,$ (6)

and since $\sigma_{1}(\Omega)$ is analytic and nonzero, then there exists
$\kappa_{1}\in\\{1,2\\}$ such that

$\sigma_{1}(\Omega)\equiv(-1)^{\kappa_{1}-1}\sigma_{1}(\Omega+2\pi
k_{1})\equiv\sigma_{1}(\Omega+2\pi k_{1}\kappa_{1}),$

and the same applies to all other singular values in the same set, i.e.

$\sigma_{i}(\Omega)\equiv(-1)^{\kappa_{1}-1}\sigma_{i}(\Omega+2\pi
k_{1})\equiv\sigma_{i}(\Omega+2\pi k_{1}\kappa_{1})$ (7)

for every $i=1,\dots,k_{1}$. As a consequence all singular values of each
$\nu$-orbit of $\tau$ has period $2\pi k_{\nu}\kappa_{\nu}$ where $k_{\nu}$ is
the multiplexing index and at the same time the size of the $\nu$-set, whereas
$\kappa_{\nu}\in\\{1,2\\}$ and it encodes the possible sign change. When
$\kappa_{\nu}=1$, we say that the singular values of the $\nu$-set are
$k_{\nu}$-multiplexed, and if additionally $\kappa_{\nu}=2$, we say that they
are signed $k_{\nu}$-multiplexed. Notice that any (signed)
$k_{\nu}$-multiplexed singular value is $2\pi k_{\nu}\kappa_{\nu}$ periodic
and thus admits a Puiseux series in $z^{1/(k_{\nu}\kappa_{\nu})}$. For a more
detailed exploration of multiplexed systems, see [26].

######  Example 3.

Consider the matrix ${\bf A}(z)$

$\begin{pmatrix}2&\sqrt{2}&\sqrt{2}&0\\\ \sqrt{2}z^{-1}&3&-1&\sqrt{2}(1+z)\\\
\sqrt{2}z^{-1}&-1&3&-\sqrt{2}(1+z)\\\
0&\sqrt{2}(1+z^{-1})&-\sqrt{2}(1+z^{-1})&4\end{pmatrix}$

whose real-variable equivalent ${\bf A}(\Omega)$ has analytic singular values
$\sigma_{1}(\Omega)=4\cos(\Omega/4)$, $\sigma_{2}(\Omega)=4\sin(\Omega/4)$,
$\sigma_{3}(\Omega)=4+4\cos(\Omega/2)$ and
$\sigma_{4}(\Omega)=4-4\cos(\Omega/2)$. Notice that

$\displaystyle|\sigma_{1}(\Omega+2\pi)|=4|\cos(\Omega/4+\pi/2)|=|\sigma_{2}(\Omega)|,$
$\displaystyle|\sigma_{2}(\Omega+2\pi)|=4|\sin(\Omega/4+\pi/2)|=|\sigma_{1}(\Omega)|,$
$\displaystyle|\sigma_{3}(\Omega+2\pi)|=|4+4\cos(\Omega/2+\pi)|=|\sigma_{4}(\Omega)|,$
$\displaystyle|\sigma_{4}(\Omega+2\pi)|=|4-4\cos(\Omega/2+\pi)|=|\sigma_{3}(\Omega)|,$

so that the resulting permutation is $\tau=(1,2)(3,4)$ and $k_{1}=k_{2}=2$.
Regarding the sign change, it is enough to test it on $\sigma_{1}$ and
$\sigma_{3}$:

$\displaystyle\sigma_{1}(\Omega+4\pi)=4\cos(\Omega/4+\pi)=-\sigma_{1}(\Omega),$
$\displaystyle\sigma_{3}(\Omega+4\pi)=4+4\cos(\Omega/2+2\pi)=\sigma_{3}(\Omega),$

showing that $\kappa_{1}=2$, $\kappa_{2}=1$. In fact $\sigma_{1}$ and
$\sigma_{2}$ have period $2\pi k_{1}\kappa_{1}=8\pi$ and are signed
$2$-multiplexed, whereas $\sigma_{3}$ and $\sigma_{4}$ have period $2\pi
k_{2}\kappa_{2}=4\pi$ and are $2$-multiplexed.

###  2.2 Periodicity of the Singular Vectors

Let us now focus on the periodicity of the singular vectors. In the previous
section we showed that each singular value has periodicity $2\pi
k_{\nu}\kappa_{\nu}$, where $k_{\nu}$ is the size of the multiplexed set and
$\kappa_{\nu}$ takes care of the possible sign change.

From [26] we know that any ${\bf A}(z)$ admits a SVD in Puiseux series with
index $L=lcm\\{k_{1}\kappa_{1},\dots,k_{R}\kappa_{R}\\}$, where $R$ is the
number of the orbits of the associated permutation. The SVD can be
reformulated with the associated real-variable matrices as ${\bf
A}(\Omega)={\bf U}(\Omega){\bf\Sigma}(\Omega){\bf V}(\Omega)^{H}$ where all
the factors have period $2\pi L$, but in general the elements of
${\bf\Sigma}(\Omega)$ have smaller periods $2\pi k_{\nu}\kappa_{\nu}$.

Here we prove that one can always take ${\bf U}(\Omega)$ such that the left
singular vectors associated to (signed) $k_{\nu}$-multiplexed singular values
are in turn $k_{\nu}$-multiplexed, i.e. if $(r,r+1,\dots,r+k_{\nu}-1)$ is the
orbit of the singular value $\sigma_{i}(\Omega)$ and ${\bf U}_{i}(\Omega)$ are
the left singular vectors in ${\bf U}(\Omega)$ associated to
$\sigma_{i}(\Omega)$, then

  * •

${\bf U}_{s}(\Omega+2\pi)={\bf U}_{s+1}(\Omega)$ for $s=r,\dots,r+k_{\nu}-1$,

  * •

${\bf U}_{r+k_{\nu}-1}(\Omega+2\pi)={\bf U}_{r}(\Omega)$,

and in particular ${\bf U}_{i}(\Omega)$ is $2\pi k_{\nu}$ periodic.

At the same time, one can always take ${\bf V}(\Omega)$ such that right
singular vectors associated to (signed) $k_{\nu}$-multiplexed singular values
are in turn (signed) $k_{\nu}$-multiplexed, i.e. if ${\bf V}_{i}(\Omega)$ are
the right singular vectors in ${\bf V}(\Omega)$ associated to
$\sigma_{i}(\Omega)$, then

  * •

${\bf V}_{s}(\Omega+2\pi)={\bf V}_{s+1}(\Omega)$ for $s=r,\dots,r+k_{\nu}-1$,

  * •

${\bf V}_{r+k_{\nu}-1}(\Omega+2\pi)\equiv(-1)^{\kappa_{1}-1}{\bf
V}_{r}(\Omega)$,

and in particular ${\bf V}_{i}(\Omega)$ is $2\pi k_{i}\kappa_{i}$ periodic.

The following proof coincide in part with Theorem 2 in [26] and Theorem 3.14
in [3]. We make use of Proposition 3.9 of [3], where it is proved that for any
para-Hermitian matrix ${\bf A}(z)$ there exists an EVD ${\bf A}(\Omega)={\bf
W}(\Omega){\bf D}(\Omega){\bf W}(\Omega)^{H}$ where the eigenvectors in ${\bf
W}(\Omega)$ have the same period as the respective eigenvalues on the diagonal
of ${\bf D}(\Omega)$, and from the proof of Theorem 3.14 in the same document,
we can also suppose that the eigenvectors ${\bf W}_{i}(\Omega)$ relative to an
eigenvalue $\lambda_{i}(\Omega)$ are $k_{\nu}$-multiplexed when the eigenvalue
itself is $k_{\nu}$-multiplexed.

######  Theorem 2.

For an analytic matrix $\textbf{A}(z)$, there exists an analytic SVD on the
unit circumference

$\textbf{A}(e^{j\Omega})=\textbf{U}(\Omega){\bf\Sigma}(\Omega)\textbf{V}(\Omega)^{H},$

where the diagonal matrix ${\bf\Sigma}(\Omega)$ may have negative entries and
contains the singular values of ${\bf A}(\Omega)$ up to the sign. Moreover,
each nonzero singular value $\sigma_{i}(\Omega)$ in ${\bf\Sigma}(\Omega)$ is
(signed) $k_{\nu}$-multiplexed and

  * •

the associated left singular vectors $\textbf{U}_{i}(\Omega)$ are
$k_{\nu}$-multiplexed, and in particular $2\pi k_{\nu}$ periodic,

  * •

the associated right singular vectors $\textbf{V}_{i}(\Omega)$ are (signed)
$k_{\nu}$-multiplexed, and in particular $2\pi k_{\nu}\kappa_{\nu}$ periodic.

###### Proof.

From Theorem 1, the SVD ${\bf A}(\Omega)=\widetilde{\bf
U}(\Omega){\bf\Sigma}(\Omega)\widetilde{\bf V}(\Omega)^{H}$ presents nonzero
singular values that are $2\pi k_{\nu}\kappa_{\nu}$ periodic depending on the
orbits of the permutation. Without loss of generality, suppose the permutation
is $\tau=(1,\dots,k_{1})(k_{1}+1,\dots)\dots$ and notice that from (7),

$\sigma_{i}(\Omega)^{2}\equiv\sigma_{i}(\Omega+2\pi k_{\nu})^{2}\quad\forall
i,$

i.e., the square of all singular values are $2\pi k_{\nu}$ periodic. Both
${\bf R}_{1}(z):={\bf A}(z){\bf A}^{P}(z)$ and ${\bf R}_{2}(z):={\bf
A}^{P}(z){\bf A}(z)$ are positive semi-definite para-Hermitian matrices and
have the same nonzero analytic eigenvalues $\lambda_{i}(\Omega)$ that are the
square of the nonzero singular values in ${\bf\Sigma}(\Omega)$. We can always
suppose, up to a sign change, that ${\bf\Sigma}(\Omega)\geq 0$ on a small
interval $\Omega\in[0,\varepsilon]$, so that if
$\sigma_{i}(\Omega)=-\sigma_{j}(\Omega)$ for all $\Omega$, then necessarily
$\sigma_{i}\equiv\sigma_{j}\equiv 0$. As a consequence, there’s a unique way
to associate the nonzero singular values $\sigma_{i}(\Omega)$ to the nonzero
eigenvalues $\lambda_{i}(\Omega)=\sigma_{i}(\Omega)^{2}$ and in particular
they have the same multiplicity, and if $\sigma_{i}(\Omega)$ is (signed)
$k_{\nu}$-multiplexed, then the eigenvalue $\lambda_{i}(\Omega)$ is
$k_{\nu}$-multiplexed with the same orbit. One can thus write down the EVD of
the para-Hermitian matrices as

$\displaystyle{\bf R}_{1}(\Omega)={\bf Q}_{1}(\Omega){\bf D}_{1}(\Omega){\bf
Q}_{1}(\Omega)^{H},$ $\displaystyle{\bf R}_{2}(\Omega)={\bf Q}_{2}(\Omega){\bf
D}_{2}(\Omega){\bf Q}_{2}(\Omega)^{H},$

where the eigenvectors in ${\bf Q}_{1}(\Omega)$ and ${\bf Q}_{2}(\Omega)$
relative to the eigenvalue $\lambda_{i}(\Omega)\not\equiv 0$ have the same
period $2\pi k_{\nu}$ and are $k_{\nu}$-multiplexed when $\lambda_{i}(\Omega)$
and $\sigma_{i}(\Omega)$ are $k_{\nu}$-multiplexed, and ${\bf D}_{1}(\Omega)$,
${\bf D}_{2}(\Omega)$ present in order the same nonzero eigenvalues on their
diagonals. From (4), we can then formulate a SVD of ${\bf A}(\Omega)$ as ${\bf
A}(\Omega)={\bf Q}_{1}(\Omega){\bf\Sigma}(\Omega){\bf\Psi}(\Omega){\bf
Q}_{2}^{H}(\Omega)$ only for the $\Omega$ where ${\bf\Sigma}(\Omega)\geq 0$,
but if some singular value is negative, we can always multiply a diagonal $\pm
1$ matrix to ${\bf\Psi}(\Omega)$ that keeps it unitary and with the same
blocks on the diagonal. Moreover, we can always suppose that the elements on
the diagonal of ${\bf\Psi}(\Omega)$ relative to identically zero singular
values are just $1$.

Let ${\bf Q}_{1}^{(i)}(\Omega)$ and $[{\bf
Q}_{2}(\Omega){\bf\Psi}(\Omega)^{H}]^{(i)}={\bf
Q}^{(i)}_{2}(\Omega){\bf\Psi}_{i}(\Omega)^{H}$ be the left and right singular
vectors relative to tho nonzero singular value $\sigma_{i}(\Omega)$. Recall
that up to a negligible set, for all $\Omega\in\mathbb{R}$ the values
$|\sigma_{i}(\Omega)|$ are distinct, so from now on we always suppose that it
holds. As a consequence, by the uniqueness of (3) and the analytic SVD ${\bf
A}(\Omega)=\widetilde{\bf U}(\Omega){\bf\Sigma}(\Omega)\widetilde{\bf
V}(\Omega)^{H}$ we have

$\displaystyle{\bf Q}_{1}^{(i)}(\Omega){\bf\Psi}_{i}(\Omega){\bf
Q}^{(i)}_{2}(\Omega)^{H}=\widetilde{\bf U}_{i}(\Omega)\widetilde{\bf
V}_{i}(\Omega)^{H}$ $\displaystyle\implies$
$\displaystyle{\bf\Psi}_{i}(\Omega)={\bf
Q}_{1}^{(i)}(\Omega)^{H}\widetilde{\bf U}_{i}(\Omega)\widetilde{\bf
V}_{i}(\Omega)^{H}{\bf Q}^{(i)}_{2}(\Omega),$

proving that each block of ${\bf\Psi}(\Omega)$, and thus the whole
${\bf\Psi}(\Omega)$, is also analytic.

We can thus consider the two unitary analytic matrices ${\bf U}(\Omega):={\bf
Q}_{1}(\Omega)$ and ${\bf V}(\Omega):={\bf
Q}_{2}(\Omega){\bf\Psi}(\Omega)^{H}$ and form the analytic SVD ${\bf
A}(\Omega)={\bf U}(\Omega){\bf\Sigma}(\Omega){\bf V}(\Omega)^{H}$ where ${\bf
U}(\Omega)$ already satisfies the thesis. Recall that the permutation is
$\tau=(1,\dots,k_{1})(k_{1}+1,\dots)\dots$, so we can focus on the first set
of multiplexed singular values, for which we can always suppose

$\displaystyle\sigma_{i+1}(\Omega)\equiv\sigma_{i}(\Omega+2\pi),$
$\displaystyle\forall i=1,\dots,k_{1}-1,$
$\displaystyle\sigma_{i}(\Omega)\equiv(-1)^{\kappa_{1}-1}\sigma_{i}(\Omega+2\pi
k_{1}),$ $\displaystyle\forall i=1,\dots,k_{1}.$

As a consequence, from the uniqueness of the decomposition (3), the $2\pi$
periodicity of ${\bf A}(\Omega)$, and the fact that ${\bf U}_{i}(\Omega)$ is
$k_{\nu}$-multiplexed we find that

$\displaystyle{\bf U}_{i+1}(\Omega){\bf V}_{i}(\Omega+2\pi)^{H}$
$\displaystyle={\bf U}_{i}(\Omega+2\pi){\bf V}_{i}(\Omega+2\pi)^{H}$
$\displaystyle={\bf U}_{i+1}(\Omega){\bf V}_{i+1}(\Omega)^{H}$
$\displaystyle\implies$ $\displaystyle\quad{\bf V}_{i}(\Omega+2\pi)={\bf
V}_{i+1}(\Omega)$

for any $i=1,\dots,k_{1}-1$ and

$\displaystyle{\bf U}_{i}(\Omega){\bf V}_{i}(\Omega+2\pi k_{1})^{H}$
$\displaystyle={\bf U}_{i}(\Omega+2\pi k_{1}){\bf V}_{i}(\Omega+2\pi
k_{1})^{H}$ $\displaystyle=(-1)^{\kappa_{1}-1}{\bf U}_{i}(\Omega){\bf
V}_{i}(\Omega)^{H}$ $\displaystyle\implies$ $\displaystyle\quad{\bf
V}_{i}(\Omega+2\pi k_{1})=(-1)^{\kappa_{1}-1}{\bf V}_{i}(\Omega)$

for any $i=1,\dots,k_{1}$. Repeating the same steps for all orbits, the thesis
is proved. ∎

######  Example 4.

Let ${\bf A}(z)$ be the same matrix as in Example 3 and consider the unitary
matrices

$\displaystyle{\bf U}(\Omega)$
$\displaystyle:=\frac{1}{2}\begin{pmatrix}\sqrt{2}&\sqrt{2}&0&0\\\
e^{-j\Omega/2}&-e^{-j\Omega/2}&1&1\\\ e^{-j\Omega/2}&-e^{-j\Omega/2}&-1&-1\\\
0&0&\sqrt{2}e^{-j\Omega/2}&-\sqrt{2}e^{-j\Omega/2}\end{pmatrix},$
$\displaystyle{\bf V}(\Omega)$
$\displaystyle:=\frac{1}{2}\begin{pmatrix}\sqrt{2}e^{j\Omega/4}&j\sqrt{2}e^{j\Omega/4}&0&0\\\
e^{-j\Omega/4}&-je^{-j\Omega/4}&1&1\\\
e^{-j\Omega/4}&-je^{-j\Omega/4}&-1&-1\\\
0&0&\sqrt{2}e^{-j\Omega/2}&-\sqrt{2}e^{-j\Omega/2}\end{pmatrix}.$

Then ${\bf A}(\Omega)={\bf U}(\Omega){\bf\Sigma}(\Omega){\bf V}(\Omega)^{H}$
and the singular values on the diagonal of ${\bf\Sigma}(\Omega)$ are in order
$\sigma_{1}(\Omega)=4\cos(\Omega/4)$, $\sigma_{2}(\Omega)=-4\sin(\Omega/4)$,
$\sigma_{3}(\Omega)=4+4\cos(\Omega/2)$ and
$\sigma_{4}(\Omega)=4-4\cos(\Omega/2)$. In this case $\sigma_{1}(\Omega)$ and
$\sigma_{2}(\Omega)$ are signed $2$-multiplexed, and in fact ${\bf
U}_{1}(\Omega)$, ${\bf U}_{2}(\Omega)$ are $2$-multiplexed since

${\bf U}_{1}(\Omega+4\pi)={\bf U}_{2}(\Omega+2\pi)={\bf U}_{1}(\Omega)$

and ${\bf V}_{1}(\Omega)$, ${\bf V}_{2}(\Omega)$ are signed $2$-multiplexed
since

${\bf V}_{1}(\Omega+4\pi)={\bf V}_{2}(\Omega+2\pi)=-{\bf V}_{1}(\Omega).$

Similarly, $\sigma_{3}(\Omega)$ and $\sigma_{4}(\Omega)$ are $2$-multiplexed,
and in fact one can check that ${\bf U}_{3}(\Omega)$, ${\bf U}_{4}(\Omega)$,
${\bf V}_{3}(\Omega)$, ${\bf V}_{4}(\Omega)$ are all $2$-multiplexed.

##  3 Diagonal Complex Decomposition

Here we discuss how to replace the signed multiplexed singular values and
vectors with non-signed ones, in fact raising the regularity of the involved
quantities by reducing their periods by a factor 2. In exchange, we have to
admit that the singular values may be complex-valued. In fact, if
$\sigma_{1}(\Omega),\dots,\sigma_{k}(\Omega)$ and the relative right singular
vectors in ${\bf V}(\Omega)$ in the decomposition of Theorem 2 are signed
$k$-multiplexed, then
$s_{i}(\Omega):=\sigma_{i}(\Omega)e^{j\Omega/(2k)}\omega_{2k}^{i-1}$ and
$\widetilde{\bf V}_{i}(\Omega):={\bf
V}_{i}(\Omega)e^{j\Omega/(2k)}\omega_{2k}^{i-1}$ are just $k$-multiplexed,
where $\omega_{p}:=e^{2\pi j/p}$ and $|s_{i}(\Omega)|=|\sigma_{i}(\Omega)|$
are the singular values of ${\bf A}(\Omega)$. As a consequence, it is
sufficient to multiply ${\bf\Sigma}(\Omega)$ and ${\bf V}(\Omega)$ by an
opportune unitary matrix to get rid of the sign change in the multiplexed
singular values and vectors.

######  Theorem 3.

For an analytic matrix $\textbf{A}(z)$, there exists an analytic complex
diagonalization on the unit circumference

$\textbf{A}(e^{j\Omega})=\textbf{U}(\Omega){\bf
S}(\Omega)\textbf{V}(\Omega)^{H},$

where the diagonal matrix ${\bf S}(\Omega)$ may have complex entries whose
absolute values are the singular values of ${\bf A}(\Omega)$ and ${\bf
U}(\Omega)$, ${\bf V}(\Omega)$ are paraunitary. Moreover, if the nonzero
singular value $\sigma_{i}(\Omega)$ is (signed) $k_{\nu}$-multiplexed, then
the relative nonzero entry $s_{i}(\Omega)$, the associated left singular
vectors $\textbf{U}_{i}(\Omega)$ and the associated right singular vectors
$\textbf{V}_{i}(\Omega)$ are all $k_{\nu}$-multiplexed, and in particular
$2\pi k_{\nu}$ periodic.

###### Proof.

Thanks to Theorem 2, there always exists an analytic decomposition

${\bf A}(\Omega)={\bf U}(\Omega){\bf\Sigma}(\Omega)\widetilde{\bf
V}(\Omega)^{H},$

where for each (signed) $k_{\nu}$-multiplexed nonzero singular value
$\sigma_{i}(\Omega)$ in ${\bf\Sigma}(\Omega)$, the associated left singular
vectors ${\bf U}_{i}(\Omega)$ are $k_{\nu}$-multiplexed, and the associated
right singular vectors $\widetilde{\bf V}_{i}(\Omega)$ are (signed)
$k_{\nu}$-multiplexed.

Let us now focus on the first set of singular values $\sigma_{i}(\Omega)$ with
$1\leq i\leq k_{1}$ that are (signed) $k_{1}$-multiplexed and define
$s_{i}(\Omega):=\sigma_{i}(\Omega)e^{j\Omega/(k_{1}\kappa_{1})}\omega_{k_{1}\kappa_{1}}^{i-1}$.
All $s_{i}(\Omega)$ are $k_{1}$-multiplexed since

$\displaystyle s_{i}(\Omega+2\pi)=$
$\displaystyle\,\sigma_{i}(\Omega+2\pi)e^{j\Omega/(k_{1}\kappa_{1})}\omega_{k_{1}\kappa_{1}}\omega_{k_{1}\kappa_{1}}^{i-1}$
$\displaystyle=$
$\displaystyle\,\sigma_{i+1}(\Omega)e^{j\Omega/(k_{1}\kappa_{1})}\omega_{k_{1}\kappa_{1}}^{i}=s_{i+1}(\Omega)$

for $i=1,\dots,k_{1}-1$ and

$\displaystyle s_{i}(\Omega+2\pi k_{1})=$
$\displaystyle\,\sigma_{i}(\Omega+2\pi
k_{1})e^{j\Omega/(k_{1}\kappa_{1})}e^{j2\pi/\kappa_{1}}\omega_{k_{1}\kappa_{1}}^{i-1}$
$\displaystyle=$
$\displaystyle\,\sigma_{i}(\Omega)e^{j\Omega/(k_{1}\kappa_{1})}\omega_{k_{1}\kappa_{1}}^{i-1}=s_{i}(\Omega)$

for $i=1,\dots,k_{1}$. Using the same reasoning, one can prove also that ${\bf
V}_{i}(\Omega):=\widetilde{\bf
V}_{i}(\Omega)e^{j\Omega/(k_{1}\kappa_{1})}\omega_{k_{1}\kappa_{1}}^{i-1}$ are
$k_{1}$-multiplexed. Repeating the same reasoning for all the (copies of all)
the orbits of $\tau$, we find a decomposition of the form

${\bf
A}(\Omega)=\sum_{i}s_{i}(\Omega){\textbf{U}}_{i}(\Omega){\textbf{V}}_{i}(\Omega)^{H}={\textbf{U}}(\Omega){\bf
S}(\Omega){\textbf{V}}(\Omega)^{H}$

with diagonal complex matrix ${\bf S}(\Omega)$. ∎

######  Example 5.

Let ${\bf A}(z)$ be

${\bf
A}(z):=\frac{1}{2}\begin{pmatrix}-\sqrt{2}j(z^{-2}+z)&-j(z+z^{-1})&-j(z+z^{-1})\\\
\sqrt{2}(z-z^{-2})&z-z^{-1}&z-z^{-1}\end{pmatrix},$

that has left and right singular vectors

$\displaystyle{\bf U}(\Omega)$
$\displaystyle:=-\frac{1}{2}\begin{pmatrix}j(e^{-j2\Omega}+e^{j\Omega/2})&j(e^{-j2\Omega}-e^{j\Omega/2})\\\
e^{-j2\Omega}-e^{j\Omega/2}&e^{-j2\Omega}+e^{j\Omega/2}\end{pmatrix},$
$\displaystyle{\bf V}(\Omega)$
$\displaystyle:=\frac{1}{2}\begin{pmatrix}\sqrt{2}e^{-j\Omega/4}&-\sqrt{2}je^{-j\Omega/4}&0\\\
e^{-j3\Omega/4}&je^{-j3\Omega/4}&\sqrt{2}\\\
e^{-j3\Omega/4}&je^{-j3\Omega/4}&-\sqrt{2}\end{pmatrix}$

and SVD ${\bf A}(\Omega)={\bf U}(\Omega){\bf\Sigma}(\Omega){\bf
V}(\Omega)^{H}$ with $\sigma_{1}(\Omega)=2\cos(\Omega/4)$,
$\sigma_{2}(\Omega)=-2\sin(\Omega/4)$. In this case $\sigma_{1}(\Omega)$ and
$\sigma_{2}(\Omega)$ are signed $2$-multiplexed, so $k_{1}=\kappa_{1}=2$.
Moreover, ${\bf U}_{1}(\Omega)$, ${\bf U}_{2}(\Omega)$ are $2$-multiplexed and
${\bf V}_{1}(\Omega)$, ${\bf V}_{2}(\Omega)$ are signed $2$-multiplexed. We
eliminate the sign ambiguity in transforming the singular values into
$s_{i}(\Omega)$ by multiplication with
$e^{j\Omega/(k_{1}\kappa_{1})}\omega_{k_{1}\kappa_{1}}^{i-1}=e^{j\Omega/4}j^{i-1}$,
thus

$s_{1}(\Omega)=\sigma_{1}(\Omega)e^{j\Omega/4}=2\cos(\Omega/4)e^{j\Omega/4},\quad
s_{2}(\Omega)=\sigma_{2}(\Omega)e^{j\Omega/4}j=-2\sin(\Omega/4)e^{j\Omega/4}j.$

Notice that they are 2-multiplexed without the sign ambiguity since

$\displaystyle s_{1}(\Omega+2\pi)$
$\displaystyle=2\cos(\Omega/4+\pi/2)e^{j\Omega/4+j\pi/2}=-2\sin(\Omega/4)e^{j\Omega/4}j=s_{2}(\Omega),$
$\displaystyle s_{2}(\Omega+2\pi)$
$\displaystyle=-2\sin(\Omega/4+\pi/2)e^{j\Omega/4+j\pi/2}j=2\cos(\Omega/4)e^{j\Omega/4}=s_{1}(\Omega).$

If ${\bf S}(\Omega)$ is a diagonal matrix with $s_{i}(\Omega)$ as diagonal
elements, then

${\bf A}(\Omega)={\bf U}(\Omega){\bf
S}(\Omega)\begin{pmatrix}e^{j\Omega/4}&&\\\ &e^{j\Omega/4}j&\\\
&&1\end{pmatrix}^{H}{\bf V}(\Omega)^{H}$

where the first two columns of the unitary matrix

${\bf V}(\Omega)\begin{pmatrix}e^{j\Omega/4}&&\\\ &e^{j\Omega/4}j&\\\
&&1\end{pmatrix}=\frac{1}{2}\begin{pmatrix}\sqrt{2}&\sqrt{2}&0\\\
e^{-j\Omega/2}&-e^{-j\Omega/2}&\sqrt{2}\\\
e^{-j\Omega/2}&-e^{-j\Omega/2}&-\sqrt{2}\end{pmatrix}$

are now $2$-multiplexed with no sign ambiguity.

##  4 Diagonal Pseudo-Circulant Decomposition

The multiplexed singular values are tightly linked to the eigenvalues of
certain structured matrices, called pseudo-circulant, that are $2\pi$ periodic
$N\times N$ normal matrices ${\bf C}(\Omega)$ for which there exist analytic
$2\pi$ periodic functions $\phi_{0}(\theta),\dots,\phi_{N-1}(\theta)$ such
that ${\bf C}(\Omega)$ is in the form

$\begin{bmatrix}\phi_{0}(\Omega)&e^{-j\Omega}\phi_{N-1}(\Omega)&\dots&e^{-j\Omega}\phi_{1}(\Omega)\\\
\phi_{1}(\Omega)&\phi_{0}(\Omega)&\ddots&\vdots\\\
\vdots&\ddots&\ddots&e^{-j\Omega}\phi_{N-1}(\Omega)\\\
\phi_{N-1}(\Omega)&\dots&\phi_{1}(\Omega)&\phi_{0}(\Omega)\end{bmatrix}$

One can prove that the eigenvalues $\lambda_{i}(\Omega)$ are all
$N$-multiplexed and all pseudo-circulant matrices are diagonalized by the same
unitary transformation

${\bf W}_{N}(\Omega):=\operatorname{diag}(e^{j\Omega k/N})_{k=0,\dots,N-1}{\bf
F}_{N}={\bf D}_{N}(\Omega){\bf F}_{N},$ (8)

where ${\bf F}_{N}$ is the $N\times N$ unitary Fourier matrix. The eigenvalues
can be expressed in term of the $\phi_{k}(\Omega)$ as

$\lambda_{i}(\Omega)=\sum_{k=0}^{N-1}e^{-j\Omega
k/N}\phi_{k}(\Omega)\omega_{N}^{-ki},$

where $\omega_{N}:=e^{j2\pi/N}$.

Moreover, any set $\\{\lambda_{1}(\Omega),\dots,\lambda_{N}(\Omega)\\}$ of
analytic $N$-multiplexed functions is the set of eigenvalues for some pseudo-
circulant matrix, i.e.

${\bf W}_{N}(\Omega)\operatorname{diag}(\lambda_{k}(\Omega))_{k=1,\dots,N}{\bf
W}_{N}(\Omega)^{H}$

is always pseudo-circulant. The functions $\phi_{k}(\Omega)$ can be expressed
in terms of the eigenvalues as

$\phi_{k}(\Omega)=\frac{1}{N}e^{j\Omega
k/N}\sum_{i=0}^{N-1}\lambda_{i}(\Omega)\omega_{N}^{ki}.$

An important technical result in [3] proves that the columns of ${\bf
W}_{N}(\Omega)$ are $N$-multiplexed, i.e.

${\bf W}_{N}(\Omega+2\pi)={\bf W}_{N}(\Omega)P_{N},$ (9)

where $P_{N}$ is the constant permutation matrix

$\begin{bmatrix}&&&1\\\ 1&&&\\\ &\ddots&&\\\ &&1&\end{bmatrix}.$ (10)

The proof of the main result showing the existence of the holomorphic pseudo-
circulant block decomposition follows the same arguments of Theorem 3.14 in
[3]. The main idea is to group the multiplexed singular values in the diagonal
complex decomposition from Theorem 3 and apply the base change ${\bf
W}(\Omega)$ to make them block pseudo-circulant. Notice that this wasn’t
possible to achieve in presence of signed multiplexed singular values. The
unitary matrices undergo the same base change, but since they are also
multiplexed thanks to Theorem 3, they also become $2\pi$-periodic.

######  Theorem 4.

For an analytic matrix $\textbf{A}(z)$, there exists an analytic decomposition

${\bf A}(z)={\bf U}(z){\bf C}(z){\bf V}(z)^{P}$

with para-unitary matrices ${\bf U}(z)$, ${\bf V}(z)$ and pseudo-circulant
block-diagonal matrix ${\bf C}(z)$, where each block has size $k_{\nu}$
corresponding to a set of (signed) $k_{\nu}$ singular values.

###### Proof.

Thanks to Theorem 3, there always exists an analytic decomposition

${\bf A}(\Omega)=\widetilde{\bf U}(\Omega){\bf S}(\Omega)\overline{\bf
V}(\Omega)^{H},$

where for each $k_{\nu}$-multiplexed nonzero singular value $s_{i}(\Omega)$ in
${\bf S}(\Omega)$, the associated left singular vectors $\widetilde{\bf
U}_{i}(\Omega)$ and the associated right singular vectors $\overline{\bf
V}_{i}(\Omega)$ are all $k_{\nu}$-multiplexed. More importantly, there are no
singular values nor singular vectors that are signed multiplexed.

If $q$ is the multiplicity of the singular value $s_{1}(\Omega)$, then there
are $q$ identical copy of the first set of singular values, and $q$ is also
the number of columns in $\overline{\bf V}_{i}(\Omega)$ and $\widetilde{\bf
U}_{i}(\Omega)$ for $i=1,\dots,k_{1}$. When reordering the repeated singular
values into forming the $q$ identical multiplexed sets, we can always suppose
that the $t$-th copy of the set correspond to the singular vectors
$\overline{\bf V}_{t,1}(\Omega)$ and $\widetilde{\bf U}_{t,1}(\Omega)$
residing in the $t$-th columns of $\overline{\bf V}_{i}(\Omega)$ and
$\widetilde{\bf U}_{i}(\Omega)$.

More specifically, If $\overline{\bf v}_{i}^{t}(\Omega)$ is the $t$-th column
of $\overline{\bf V}_{i}(\Omega)$, then $\overline{\bf
V}_{t,1}(\Omega):=[\overline{\bf v}_{1}^{t}(\Omega),\dots,\overline{\bf
v}^{t}_{k_{1}}(\Omega)]$ and analogously with $\widetilde{\bf
u}_{i}^{t}(\Omega)$, $\widetilde{\bf U}_{i}(\Omega)$ and $\widetilde{\bf
U}_{t,1}(\Omega)=[\widetilde{\bf u}_{1}^{t}(\Omega),\dots,\widetilde{\bf
u}_{k_{1}}^{t}(\Omega)]$. The indexes $t,1$ indicate that among the singular
values in the first $k_{1}$-multiplexed orbit of the permutation $\tau$, we
are investigating the $t$-th copy. Do the same for $\widetilde{\bf
V}_{t,1}(\Omega)$ from $\widetilde{\bf V}_{i}(\Omega)$ and notice that

$\displaystyle\overline{\bf V}_{t,1}(\Omega)$
$\displaystyle=e^{j\Omega/(k_{1}\kappa_{1})}\widetilde{\bf
V}_{t,1}(\Omega)\operatorname{diag}(\omega_{k_{1}\kappa_{1}}^{i})_{i=0,\dots,k_{1}-1}$
$\displaystyle=e^{j\Omega/(k_{1}\kappa_{1})}\widetilde{\bf
V}_{t,1}(\Omega){\bf D}_{k_{1}}(-2\pi/\kappa_{i}),$

where ${\bf D}(\Omega)$ is defined in (8).

By hypothesis, the columns of $\overline{\bf V}_{t,1}(\Omega)$ and
$\widetilde{\bf U}_{t,1}(\Omega)$ are $k_{1}$-multiplexed for any $t$, so
$\widetilde{\bf U}_{t,1}(\Omega+2\pi)=\widetilde{\bf U}_{t,1}(\Omega){\bf
P}_{k_{1}}$ and the same with $\overline{\bf V}_{t,1}(\Omega)$ where ${\bf
P}_{k_{1}}$ is the permutation matrix in (10). Focusing on $t=1$, and
recalling that also ${\bf W}_{k_{1}}(\Omega)$ is $k_{1}$-multiplexed, we have

$\displaystyle\widetilde{\bf U}_{t,1}(\Omega){\bf W}_{k_{1}}(\Omega)^{H}$
$\displaystyle=\widetilde{\bf U}_{t,1}(\Omega){\bf P}_{k_{1}}{\bf
P}_{k_{1}}^{H}{\bf W}_{k_{1}}(\Omega)^{H}$ $\displaystyle=\widetilde{\bf
U}_{t,1}(\Omega+2\pi){\bf W}_{k_{1}}(\Omega+2\pi)^{H},$

and the same applies with $\overline{\bf V}_{t,1}(\Omega)$ instead of
$\widetilde{\textbf{U}}_{t,1}(\Omega)$. As a consequence, we can define the
matrices $\textbf{U}_{t,1}(\Omega):=\widetilde{\bf U}_{t,1}(\Omega){\bf
W}_{k_{1}}(\Omega)^{H}$ and
$\textbf{V}_{t,1}(\Omega):=\overline{\textbf{V}}_{t,1}(\Omega){\bf
W}_{k_{1}}(\Omega)^{H}$ that are both analytic and $2\pi$ periodic and with
orthonormal columns. If now
${\bf\Sigma}_{1}(\Omega):=\text{diag}(\sigma_{i}(\Omega))_{i=1,\dots,k_{1}}$and
${\bf\Delta}_{1}(z):=\text{diag}(s_{i}(\Omega))_{i=1,\dots,k_{1}}$, then

$\displaystyle\widetilde{\bf U}_{t,1}(\Omega)$
$\displaystyle{\bf\Delta}_{1}(\Omega)\overline{\bf V}_{t,1}(\Omega)^{H}$
$\displaystyle=\widetilde{\bf U}_{t,1}(\Omega){\bf W}_{k_{1}}(\Omega)^{H}{\bf
W}_{k_{1}}(\Omega){\bf\Delta}_{1}(\Omega){\bf W}_{k_{1}}(\Omega)^{H}{\bf
W}_{k_{1}}(\Omega)\overline{\bf V}_{t,1}(\Omega)^{H}$ $\displaystyle={\bf
U}_{t,1}(\Omega){\bf C}_{1}(\Omega){\bf V}_{t,1}(\Omega)^{H},$

where $\textbf{C}_{1}(\Omega):={\bf
W}_{k_{1}}(\Omega){\bf\Delta}_{1}(\Omega){\bf W}_{k_{1}}(\Omega)^{H}$ is a
$2\pi$ periodic analytic pseudo-circulant matrix relative to the first set of
singular values, since ${\bf\Delta}_{1}(\Omega)$ has the diagonal elements
that are $k_{1}$-multiplexed. Repeating the same reasoning for all the copies
of all the orbits of $\tau$, we find a decomposition of the form

${\bf A}(z)=\sum_{t,i}{\textbf{U}}_{t,i}(z){\bf
C}_{i}(z){\textbf{V}}_{t,i}^{P}(z)={\textbf{U}}(z){\bf
C}(z){\textbf{V}}^{P}(z)$

with pseudo-circulant block-diagonal matrix ${\bf C}(z)$. ∎

In a sense, the pseudo-circulant decomposition tells us that the only causes
of non-holomorphicity in the SVD is given separately by each multiplexed set
of singular values, and they can be removed simultaneously from the singular
vectors and the singular values by allowing a block diagonal middle matrix
with specially structured blocks. Moreover, the possibly signed singular
values and vectors can be transformed nonetheless into a pseudo-circulant
structure.

######  Example 6.

Let ${\bf A}(z)$ be as in Example 5. We know that ${\bf A}(\Omega)={\bf
U}(\Omega){\bf S}(\Omega){\bf V}(\Omega)^{H}$ with

$\displaystyle{\bf U}(\Omega)$
$\displaystyle:=-\frac{1}{2}\begin{pmatrix}j(e^{-j2\Omega}+e^{j\Omega/2})&j(e^{-j2\Omega}-e^{j\Omega/2})\\\
e^{-j2\Omega}-e^{j\Omega/2}&e^{-j2\Omega}+e^{j\Omega/2}\end{pmatrix},$
$\displaystyle{\bf S}(\Omega)$
$\displaystyle:=\begin{pmatrix}2\cos(\Omega/4)e^{j\Omega/4}&0&0\\\
0&-2j\sin(\Omega/4)e^{j\Omega/4}&0\end{pmatrix},$ $\displaystyle{\bf
V}(\Omega)$ $\displaystyle:=\frac{1}{2}\begin{pmatrix}\sqrt{2}&\sqrt{2}&0\\\
e^{-j\Omega/2}&-e^{-j\Omega/2}&\sqrt{2}\\\
e^{-j\Omega/2}&-e^{-j\Omega/2}&-\sqrt{2}\end{pmatrix},$

where all the $s_{i}(\Omega)$ and the columns ${\bf U}_{1}(\Omega)$, ${\bf
U}_{2}(\Omega)$, ${\bf V}_{1}(\Omega)$, ${\bf V}_{2}(\Omega)$ are
$2$-multiplexed. Since the diagonal entries of ${\bf S}(\Omega)$ are
$2$-multiplexed, the base change ${\bf W}_{2}(\Omega)$ in (8) transforms it
into

${\bf C}(z):={\bf W}_{2}(\Omega){\bf S}(\Omega)\begin{pmatrix}{\bf
W}_{2}(\Omega)^{H}&\\\ &1\end{pmatrix}=\begin{pmatrix}1&1&0\\\
z&1&0\end{pmatrix},$

that is block pseudo-circulant with the only pseudo-circulant block relative
to the functions $\phi_{0}(z)=1$, $\phi_{1}(z)=z$. As a consequence, one can
find the holomorphic pseudo-circulant decomposition as ${\bf A}(z)={\bf
U}^{\prime}(z){\bf C}(z){\bf V}^{\prime}(z)^{H}$ with

$\displaystyle{\bf U}^{\prime}(z)$ $\displaystyle:={\bf U}(z){\bf
W}_{2}(z)^{P}=\frac{1}{\sqrt{2}}\begin{pmatrix}-jz^{-2}&-j\\\
-z^{-2}&1\end{pmatrix},$ $\displaystyle{\bf V}^{\prime}(z)$
$\displaystyle:={\bf V}(z)\begin{pmatrix}{\bf W}_{2}(z)^{P}&\\\
&1\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}\sqrt{2}&&\\\ &z^{-1}&1\\\
&z^{-1}&-1\end{pmatrix}.$

##  5 Conclusions

We have studied three different decompositions for possibly rectangular
matrices ${\bf A}(z)$ that are analytic at least on $S^{1}$. From the known
SVD that is analytic with respect to $z^{1/L}$, but with possibly negative
singular values, we have shown how to reduce the parameter $L$ separately for
every singular value, first by halving it by removing the sign change but
relaxing the singular values in the complex plane, and then by transforming
the multiplexed singular subspaces into an holomorphic pseudo-circulant
structure. This transformation shows that sign ambiguity plays no role in a
possible decoupling of different multiplexed systems, no matter how they have
been interlaced, akin to the eigenvalue decomposition of para-Hermitian
matrices.

Future work will focus on the design of algorithms for the computation of such
decompositions, in particular how to retrieve the pseudo-circulant
factorization without having to compute beforehand the singular values.
Another feature to be yet studied is the stability of all these
factorizations, whose analysis will be based on the preliminary results
already shown in [3] about the stability of the eigenvalues of para-Hermitian
matrices.

## Acknowledgements

The author thanks professor Stephan Weiss for its help and insight on the
topic and the presentation of the paper.

This work was partially supported by Alfred Kordelinin säätiö Grant No. 210122
and by the European Union ERC consolidator grant, eLinoR, no 101085607.

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