# Modal analysis of the scattering coefficients of an open cavity in a waveguide

###### Abstract

The characteristics of an acoustic scatterer are often described by scattering coefficients. The understanding of the mechanisms involved in the frequency dependent features of the coefficients has been a challenge task, owing to the complicated coupling between the waves in open space and the modes inside the finite scatterer. In this paper, a frequency-dependent modal description of the scattering coefficient is utilized to study the modal properties of the scatterer. The important role that eigenmodes play in defining the features of the scattering coefficients is revealed via an expansion of the coefficients by the eigenmodes. The results show the local extrema of the scattering coefficients can be attributed to the constructive/destructive interference of resonant and non-resonant modes. In particular, an approximated equation, which is equivalent to the standard Fano formula, is obtained to describe the sharp anti-symmetric Fano characteristics of the scattering coefficients. The special cases where scattering is dominated by a single resonance eigenmode, corresponding to the “resonance transmission”, are also illustrated.

###### keywords:

Open cavity; Scattering coefficient; Eigenmode; Fano resonance^{†}

^{†}journal: Wave Motion

## 1 Introduction

The scattering coefficients of acoustic scatterers are important parameters as they are used to relate incident and scattered waves. Traditionally, sound scattering in a duct by a muffler is described by one-dimensional (1D) transmission line theory, and explained by the mismatch of the specific impedance at the inlet and outlet of themuffler Kinsler2000 . To accommodate cross modes in a duct with scatterers, the wave matching technique selamet1999acoustic ; kirby2005point is used. The finite element and boundary element methods, are used to determine the scattering coefficients when the geometries of the ducts and scatterers are complicated graf2013determination ; wang2014impedance . While these wave matching and numerical methods are capable of producing accurate coefficients, they are less useful in directly delivering the physical insight into the peaks and valleys in the coefficient curves.

An alternative approach, which is motivated by the obervation of trapped
and quasi-normal modes inside the scatterers, is to describe the scattering
in terms of the coupling between the waves in the ducts and modes
inside the scatterer. This was inspired by the early work of Flax
*et al.* flax1978theory , where the sound scattering from
submerged elastic bodies is affected by various kinds of interference
between the resonance scattering at the eigenfrequencies of the vibration
of the body and the rigid-body scattering. Recently, such approach
has been used to explain the peaks and valleys in the transmission
loss curve of an expansion chamber subject to an incident plane-wave
in a 1D duct pan2015QuasiNormalModes . It was demonstrated
that the characteristics (complex eignevalues and mode shape functions)
of the frequency-dependent quasi-normal modes of the expansion chamber
kergomard2006resonance allow the correct expansion of the
sound pressure in the chamber. The coupling of the quasi-normal modes
with the incident and transmitted sound waves sheds some light on
the transmission loss. For example, the minimum values in the transmission
loss occur when the frequency of the incident wave equals the real
part of the eigenfrequency of the quasi-normal mode. On the other
hand, the maximum transmission loss appears at those frequencies where
the superimposed contribution from the participated modal factors
is at minimum. However, the extension of these previous works to two-dimensional
(2D) or three-dimensional (3D) expansion chambers and modelling of
the coupling between the quasi-modes in the chambers and the incident
and transmitted waves including cross modes, is not straight forward.
Furthermore, because of the involvement of cross mode components in
the 2D and 3D configurations, extra coupling mechanisms due to participation
of the cross modes may lead to a more complete understanding of the
characteristics of the scattering coefficients.

Most recently, Maksimov *et al*. maksimov2015coupled
and Lyapina *et al*. lyapina2015bound looked at the aforementioned
scattering problem from a more fundamental view of the acoustical
properties of an open cavity. An open cavity is a finite acoustical
space with defined boundary conditions at its internal wall (*e.g.,
a* rigid-wall condition) and some boundary areas open to infinite
space(s), such as semi-infinite waveguides. Because of the energy
exchange between the sound fields inside the open-cavity and the infinite
space(s), the sound field inside the cavity is characterized by the
non-Hermitian Hamiltonian. They used the acoustic coupled mode theory
for the calculation of the frequency-independent eigensolutions of
the open cavity and revealed a mechanism resulting in acoustic trapped
modes in the cavity, namely Friedrich–Wintgen two-mode
full destructive interference. Although being used to explain the
occurrence of Fano resonance, the nonorthorgonal properties and incompleteness
of the frequency-independent eigensolutions of the open cavity make
the direct modal interpretation of the transmission coefficient (which
is a forced scattering problem at the frequency of an incident wave)
a difficult task. Xiong *et al*. xiong2016fano , on the
other hand, derived the scattering matrix of the open cavities due
to incident waves from one of the waveguide connected to the cavity.
Frequency-dependent eigensolutions of the effective Hamiltonian matrix
(including the interaction between cavity and connected waveguides)
for the sound field in the cavity were used to describe the scattering
coefficients and to explain the links between the eigenmodes and a
trapped mode and the corresponding transmission zero. Their contribution
makes possible an analysis of the scattering coefficients of the open
cavities by using the frequency-dependent eigensolutions of the cavity.

In this paper, the method for frequency-dependent eigenmodes and the
scattering matrix developed by Xiong *et al*. xiong2016fano
is adopted. Instead of focusing on the Fano resonance induced by trapped
modes, it is used to calculate and explain the general scattering
features of the open cavities connected with waveguides (*e.g.,*
conventional muffler configuration) and the roles played by eigenmodes
in determining the frequency-dependent features of the scattering
coefficients. Through numerical studies, it is revealed that extrema
in scattering coefficients are *generally* a result of interference
between eigenmodes, rather than the contribution from single resonant
mode, as is traditionally assumed. The Fano resonance induced by highly
localized modes (quasi-trapped modes), as observed by Hein *et al.* hein2012trapped ,
is also revisited in terms of frequency-dependent eigenmodes. Finally,
some remarks are made to clarify the usage of frequency-dependent
and frequency-independent modes in conducting modal analysis in scattering
problems.

## 2 Modal description of the scattering coefficients

This paper considers the scattering problem in a 2D acoustic scatterer, comprising a cavity connected by uniform ducts. Omitting the time-dependence term, , the sound pressure field is governed by Helmholtz equation.

(1) |

and corresponding boundary conditions, where is the wavenumber.
Since (where is the speed of sound),
will be used hereafter to represent source frequency, *i.e.*,
whenever “frequency” is mentioned, it refers to the wavenumber.
For the sake of simplicity, the rigid-wall boundary condition is assumed
for the cavity and ducts. Although details of the coupled mode theory
maksimov2015coupled ; lyapina2015bound ; xiong2016fano have already
been published, this paper provides a brief description of the derivation
using coupled mode theory for the scattering coefficients of the scatterer
in terms of frequency-dependent eigenmodes, for the convenience of
the reader and to make the paper self-contained.

The first step is to express the sound pressure in terms of a local
basis in different regions. The geometry is partitioned into
regions: a closed cavity and *N* semi-infinite
ducts (). The pressure field in the
duct is expanded into duct modes when taking a local coordinate
with and axes that are, respectively, perpendicular and
vertical to the duct-cavity interface:

(2) |

where and are respectively, the amplitude for
the *incident* and *scattered* modes. The transverse
functions are given by
where is the width of the duct. Finally,
is the axial wavenumber such that,

The pressure field in the cavity may be expanded by closed cavity modes such that

(3) |

where is the *closed* cavity mode satisfying

(4) |

with Numann (rigid) boundary conditions at the duct-cavity interfaces and at other boundary surface of the cavity, where is the eigenvalue.

Although does not satisfy the continuity condition for particle velocity at the cavity-duct interfaces, it is worth emphasizing that the expansion in Eq. (3) is valid as the matching conditions for particle velocity must not be imposed directly, which is a consequence of the fact that the convergence in Hilbert space does not imply point-wise convergence at the boundary viviescas2003openopticalcavities ; jackson1999classical .

In a third step, a key equation is derived to determine the coefficient for the cavity upon incident waves by applying the continuity condition of pressure and velocity in a limiting sense infinitesimally close to the boundary:

(5) |

where is the identity matrix,
is an unknown column vector containing the coefficients .
The column vector contains the coupling constant
between the rigid cavity mode and the
duct mode maksimov2015coupled ; pichugin2001openbiliiards ,*
i.e.*, where the
integral is evaluated over the interfaces between the cavity and the
duct, and is the corresponding surface element.
The matrix has the following form:

(6) |

where is a diagonal matrix containing the squared eigenfrequencies , and symbol denotes the conjugate transpose. It then follows that the scattered wave (the outgoing duct mode in the duct) is found as

(7) |

The scattering coefficients are defined by . Substituting the solution of Eq. (5) into Eq. (7) yields

(8) |

which relates the scattered wave to the incident wave .

.

In a fourth step, an eigenvalue problem (EVP) of subject to the source frequency is solved as

(9) |

where and are respectively the frequency-dependent eigenvalue and eigenvector. Since is a non-Hermitian matrix, its eigenvectors satisfy the bi-orthogonal relation (see Appendix),

(10) |

where the superscript denotes a transpose. Alternatively, Eq. (10) may be written as

(11) |

where is the mode-shape function corresponding to . Note that the integral is evaluated over , the domain occupied by the cavity,.

Finally, expanding in Eq. (5)
in terms of eigenvectors of ,* i.e.*,
and substituting it into Eq. (7) yields the
expression for scattering coefficients in terms of eigenmodes

(12) |

where and .

Compared with Eq. (8), Eq. (12) expresses the scattering coefficients explicitly as a superposition of individual eigenmode components. The first term on the right-hand side of Eq. (12)comes from the direct reflection, while each term in the summation corresponds to scattering by eigenmode . The term measures the coupling between eigenmode and the incident duct mode . Similarly, provides a measurement of the coupling between eigenmode and the scattered duct mode .

## 3 Results and discussions

The aim of this section is to demonstrate the role that the eigenmodes of the acoustic scatterer play in sound scattering, which is a question originally put forward by yu2016acoustic in their analysis of wave propagation in continuous right-angled bends. This question can now be answered mathematically using Eq. (12), by showing that scattering coefficients can indeed be expressed in terms of the eigenmodes of an acoustic scatterer. Numerical results will be used to support this answer.

In Eq. (12), the summation terms are characterized by the inverse of . One might therefore assume that is dominated by each eigenmodes within each frequency range and that achieves extreme values. The numerical study, however, will show that the local extrema of the scattering coefficients, are generally the result of constructive or destructive interference of eigenmodes, rather than by dominating resonance alone as one would expect intuitively. In the following parts of this section, the typical interference of eigenmodes will be covered first in Sec. 3.1. Then Sec. 3.2 will discuss a special narrow band interference known as Fano Resonance. Section 3.3 will be devoted to the phenomenon of resonance transmission.

It is worth noting that, although the theory proposed in Sec. 2 is valid for scatterers of arbitrary-shape and with multiple connecting ducts, for the sake of simplicity, the numerical study will be restricted to scatterers consisting a rectangular cavity connected by two uniform ducts, with the geometries illustrated in Fig. 2. The scatterer with two connecting ducts is the simplest yet sufficiently representative of its kind, whereas a rectangular cavity enables an analytical solution of the closed cavity modes ; otherwise, the complexity of mode classification and subsequent analysis may mask important physical phenomena.

In the following context, symbols L and R are used to denote ducts on the left- and right-hand sides, respectively. The scattering coefficients, owing to the symmetry of the systems, are reduced to the transmission and reflection coefficients, and , respectively. For simplicity, only is examined here. Using Eq. (12) and modal decomposition by each eigenmode , is expressed as

(13) |

where

(14) |

and is the index of the bi-orthogonal eigenmodes. For a rectangular cavity where variable separation is possible, is represented by a pair of non-negative values . The analysis of is omitted as it can be conducted following the same procedures.

### 3.1 Typical case: interference of eigenmodes

The first to be considered is a short expansion cavity, depicted by Fig. 2(b), which supports a typical case of interference for eigenmodes. The system parameters are m, m and m, and an adjustable vertical offset is used to dictate the geometrical symmetry about the axis ( is considered here). Since the length of the cavity, , is comparable with its width, , one can anticipate that the 2D modes with non-uniform transverse functions may play an important role in sound scattering.

Figure 3(a) presents the amplitude of versus frequency.
Several peaks are observed at and
and several dips at and .
Using Eq. (13), one can decompose
into a contribution by individual eigenmodes. Due to the transversal
symmetry of the system, only transversally symmetric modes can participate
in the scattering of the plane wave, *i.e.*, only the (0,0),
(0,2) and (1,0) modes need to be taken into account for .
Figure. 3(d) and (e) show , which are the eigenvalues of
the corresponding modes. It is noted that (
) varies slowly with *k* except for , while
always start from 0 when and decreases rapidly
as *k* increases. Figure 3(d) shows the components of ,
*i.e.*, (), as functions
of *k*. When *k* approaches , the modal transmission
coefficient of the mode experiences resonance, leading
to the resonance peak of each , *e.g.*, at
for , at for and
at for . The width of each
resonance peak, on the other hand, is dictated by ,
*e.g.*, has a wide peak as
grows large, while the peak of is relatively narrow
as grows small.

A comparison between Fig 3(a) and Fig. 3(b) indicates that there are no direct relations between the peaks of and the extrema of (except for the first peak at , which will be discussed in Sec. 3.3). Instead, the local minimum in at is due to the destructive interference between the (0,2), (0,0) and (1,0) modes. The local maximum at arises from the constructive interference between (0,2), (1,0), and (0,0) eigenmodes, whereas the destructive interference between (0,2) and (1,0) gives rise to the local minimum of at .

Figure 4(a) depicts the amplitude of the transmission coefficient
of the waveguide mode, , for a long cavity (
m, m and m), as shown in Fig. 2(a). Within the
frequency range of interest (), the components
of , *i.e.*, are mainly due to the
first five eigen-modes () of the
open cavity. Figure 4(b) shows that are also characterized
by their resonance peaks. The comparison between Fig. 3(d) and Fig.
4(g) suggests that the local extrema in of the long cavity
are also due to the superposition of eigenmodes. For example, at ,
the maximum of is not caused by the resonance peak of
alone, while and
also have non-negligible contributions to the transmission coefficient.
However, the latter two terms cancelled each other leaving the (1,0)
eigenmode of the cavity as the sole contributor to at frequencies
around . A similar analysis can be made
for the other peaks, except for the first one at ,
which is a case of resonance transmission (to be discussed in Sec.
3.3). The dips in , on the other hand, are due to the
destructive interference of the eigenmodes.

### 3.2 Fano resonance

The eigenmodes responsible for the transmission coefficients in Figs. 3 and 4 have relatively large imaginary parts. The open cavity also has another type of eigenmodes, known as quasi-trapped modes, whose eigenvalues have very small imaginary part and hence result in very narrow resonance peaks. Although the involvement of quasi-trapped modes in producing the sharp asymmetric Fano resonance is well known hein2010fano ; hein2012trapped ; xiong2016fano , the theoretical analysis in Sec. 2. will be used to explain how these quasi-trapped modes are involved in producing the Fano resonances.

Consider again the short cavity with parameters of m, m, m, and m. For this case the quasi-trapped modes emerge as an offset is introduced to make the cavity asymmetric about the central axis hein2012trapped . Figure 5(a) presents the amplitude of versus . Compared with Fig. 3(a) , it is found that extra peaks and dips emerge around and in Fig 5(a), showing typical Fano resonances. To explain the formation of the Fano resonances, the contribution to of each individual mode is plotted in Fig. 5(b). Due to the broken transversal symmetry, the original fully localized modes (0,1) and (0,3) modes become quasi trapped modes with and , now participate in the coupling with the waveguide waves. As shown in Figs 5(b) and (c), that and now have very small imaginary parts. In Fig. 5(b), and display sharp resonance behaviors; the corresponding sharp phase changes are also observed in Fig. 5(c) at the peak frequencies.

An interpretation of this phenomenon is straightforward following
the wave superposition expression in Eq. (12).
Taking the first Fano resonance ( )
as an example, in the vicinity of , the
quasi-trapped mode (0,1) demonstrates a resonance response, and hence
has a sharp phase change, while the responses of
the other modes vary slowly in terms of amplitude and phase. Therefore,
below and above the resonance frequency of , constructive
and destructive interferences between and components
of other eigenmodes give rise to a local maximum and minimum, respectively,
of the . The contribution by non-resonance modes can be
approximated by a complex constant. As a result, the Fano resonance
can be characterized as the superposition of a resonantly excited
eigenmode (here, ) and background non-resonance
response , *i.e.*,

(15) | |||||

where in the third row, the numerator of the first term, originally expressed as in Eq. (12), is approximated by a constant , and in the denominator by a constant . The approximations are valid for the narrow bandwidth in which Fano resonance occurs, as the approximated terms vary slowly with frequency. The frequency-dependent can be taken as a constant , as is almost a constant within the frequency band.

On the other hand, it is well known that the standard Fano formula
miroshnichenko2010fano in terms of sound power transmission,
in *the vicinity* of can be written as:

(16) |

where is the reduced resonance frequency, measures the asymmetry of the resonance shape, and is the normalization constant.

A way to verify Eq. (15) is to fit the to the actual sound power transmission curve to determine the value of and check if it agrees with by eigensolution (Eq. (9)) within the narrow band. It is shown in Fig. 6 the original and fitted curve of using Eq. (15) in Fig. 6., Excellent agreement is observed between the actual sound power transmission curve and the fitted curve using Eq. (15), while the fitted parameter is also very close to the frequency-dependent eigen-value at the resonance frequency for of the (0,1) eigenmode, as is shown in Table. 1. The results of curve fitting indicate that Eq. (15) indeed serves as a good description of the Fano resonance caused by the quasi trapped mode (0,1).

A further observation, according to Fig. (6) and Table. 1, is that the fitted curve using Eq. (15) agrees well with that using the standard Fano formula in Eq. (16). It is interesting that the two fitted curves exhibit excellent agreement not only within the narrow band where Fano resonance occurs, but at other frequencies as well. It is an indicator that Eq. (15) is actually equivalent to the standard Fano formula - they cannot agree globally unless they are describing the same function of , or more exlicitly, they agree globally because they are equivalent and are describing the same function of .

### 3.3 Resonance transmission

The first local maximum in in Figs. 3(a), 4(a) and 5(a), at , is contributed by the resonance peak of only , while the components of all other eigenmodes are negligibly small. In other words, the transmission peak at in is dictated by the resonance of mode alone, and hence is called “resonance transmission” to distinguish it from the cases discussed in Sec. 3.1 and 3.2, where destructive and constructive superposition of eigenmodes led to local extrema in the scattering coefficients. The condition for the occurrence of resonance transmission is very strict in that it requires the scattering to be dominated by individual modes. In some cases, these conditions are fulfilled such that resonance transmission can be observed. It is considered the scattering coefficient for the same asymmetrical short cavity discussed in Sec 3.2. Figures 7(a) and (b) present respectively the amplitude of and the modal components , and the comparison of these indicates resonance transmission at and .

At the first resonance transmission ( ),
is almost dominated by , the modal component
by () eigen-mode. Figures. 7(d) and
(e) demonstrate that the modal coupling factors and
exert a crucial influence, as generally only when both
of them take relatively large values can play an important
role in . For example, correponds to a
small and a large , leading to
an ignorable transmitted wave owing to the poor coupling
between the eigenmodes and the transmitted first order duct
modes, despite the fact that the eigenmode is sufficiently
excited by the incident plane wave . A similar reasoning
applies to , where the (0,2) eigenmode has a negligible
contribution to owing to the small ,
*i.e.*, the poor coupling between the (0,2) modes and the transmitted
first order duct mode.

The above observation, however, does not hold for the squasi trapped mode (0,1). Although is small (owing to weak coupling between the eigenmode and the incident plane wave ) , the component of from the mode is still large. It is reasoned that the eigenvalue of the quasi trapped mode is almost real, giving in Eq. (14), and hence the term very large value when is close to . Therefore, in such a situation, only the (0,1) eigenmodes contributes significantly to the at , where achieves its local maxima. In the same way, it is possible to show that the resonance transmission close to is dominated by (0,3) mode.

It is noted that, in Fig. 7(a) is sometimes greater than 1. For the peak around , takes a value of around 1.8, however, it does not correspond to any transmitted sound power as the first-order mode is evanescent and carrying zero net energy flux when . At frequencies around , the axial wavenumber is very small, meaning that the transmitted sound power carried by the first order duct mode is actually less than that carried by the incident zeroth-order duct mode, even when is greater than unity.

## 4 Further discussion about Eq. (9)

In Eq. (12), the bi-orthogonal eigenmodes
defined by Eq. (9) are used to decompose the scattering
coefficients, and have been employed throughout this paper. They are
*frequency-dependent*. It is noted that Eq. (8)
can also be used to define another type of EVP, which gives rise to
the *frequency-independent* eigenmodes, satisfying,:

(17) |

where is defined by Eq. (6) but is a function of eigenvalue .

The solution of the frequency-independent EVP requires

(18) |

As Eq. (8) includes the inverse of , it suggests that eigenvalues of Eq. (17) are exactly the poles of (and, in general, the scattering matrix). In general, takes complex value, and is known as the complex resonance frequency of the scatterer, with being the resonance frequency and characterizing the decay of the mode. For some special cases, takes a real value, corresponding to localized mode with no radiation loss , which is usually referred to as a trapped mode or bound state in the continuum lyapina2015bound ; linton2007embeddedtrappedmode ; xiong2016fano .

As defined, the frequency-dependent EVP yields the bi-orthogonal eigenmodes for the forced response of the scatterer, whilst the frequency-independent EVP produces the modal characteristics of the free vibration of the system. As the response of the scatterer to an incident wave is a forced vibration problem, the use of the frequency-dependent eigenmodes is appropriate.

However, in some previous research, frequency-independent eigenmodes were sometimes adopted to deal with the forced response of the system hein2012trapped ; pelat2009use ; yang2013mechanism ; yu2016acoustic . Owing to the nonlinearity of Eq. (17), and hence the absence of orthogonality and completeness of as a complete set, the modal expansion of the sound field by the frequency-independent bases lacks justification.

A typical issue encountered when the frequency-independent eigenmodes are used for a forced response problem can be found in Ref. yu2016acoustic , where some peaks/dips in transmission loss (TL) at continuous right-angled corners were attributed to even/odd distributed frequency-independent eigenmodes. An analysis following the procedures in the present paper, however, shows that the extrema in the TL are, as expected, the result of constructive/destructive interference of frequency-dependent eigenmodes, rather than the product of single eigenmodes. Examples of this issue can also be found in the modelling of sound propagation in street canyons using non-orthogonal frequency-independent eigenmodes pelat2009use and in the expansion of the sound field in and outside a parallel sound barrier by frequency-independent eigenmodes yang2013mechanism .

However, the frequency-independent eigensolutions may be used for
the forced response of the scatterer in some special cases when (quasi-)
trapped modes are involved. Hein *et. al.*, hein2012trapped ,
employed a finite element solver to tackle complex resonance for the
investigation of quasi-trapped modes and corresponding Fano-resonance
in duct-cavity system. It is reasonable to do so because within the
narrow band of quasi-trapped modes, Eq. (15)
can be further approximated to (readers may refer to the discussion
in Sec. 3.2)

(19) |

by noting the fact from Fig. 8 that frequency-dependent eigenvalue can be well approximated by frequency-independent eigenvalue . Hence, Eq. (19) suggests that , the frequency-independent eigensolution is able to characterize the frequency dependency of scattering coefficient in such a special case.

## 5 Conclusions

In this paper, the scattering coefficients of open cavities in an acoustical waveguide were investigated via analysis of eigenmodes. By utilizing the bi-orthogonal frequency-dependent eigenmodes, the scattering coefficients can be explicitly expressed as the superposition of eigenmodes, allowing for a study of the roles that eigenmodes play in acoustic scattering.

It was shown that the local extrema in the scattering coefficients generally result from the constructive and destructive interference of the resonance and non-resonance modes. In particular, an approximation equation was derived to describe the sharp asymmetric line-shape of scattering coefficients within the narrow-band of Fano resonance induced by quasi-trapped modes. The equivalence of the approximation equation and the standard Fano formula was demonstrated. In addition, it was demonstrated that, in some special cases, the sound scattering coefficient is dominated by single modes, giving rise to resonance transmission

The theoretical analysis was limited to two-dimensional acoustic resonators, and the numerical study was restricted to resonators with simple rectangular cavities. However, it is straightforward to extend the analysis to three-dimensional irregular resonators, where more complicated phenomena may be encountered in the analysis of the frequency-dependent features of of scattering coefficients.

## Acknowledgements

The financial support from the Australian Research Council (ARC LP) is gratefully acknowledged. The first author is also grateful for the sponsorship from the China Scholarship Council.

## Appendix A Bi-orthogonal relation in Eq. (10)

Let and be the eigenvalue and eigenvector, respectively, of the non-Hermitian matrix such that

(20) |

where subscript is for labelling and generally take a complex value.

Similarly, let and be the eigenvalue
and eigenvector of the adjoint matrix of , *i.e.*,

(21) |

where since is non-Hermitian.

Taking the adjoint of Eq. (21) yields:

(22) |

By comparing Eq. (20) and Eq. (22), one can derive the following correspondence from the uniqueness of the eigenspectrum of :

(23) |

Right multiplying to the above equation and left multiplying to Eq. (20) (where is replaced by ) and subtracting the resulting equations gives,

(24) |

such that the case holds
unless , *i.e.*:

(25) |

With proper normalization, Eq. (25) gives the normalized bi-orthogonal relation

(26) |

which relates the left and right eigenvectors of a non-Hermitian matrix .

Furthermore, if is symmetric, then the transpose of Eq. (20) is

(28) |

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