Diagonalisation of Para-hermitian Matrix :is it always possible ?

Sylvie Icart

Laboratoire I3SUniversite Cote d’Azur – Universite Nice Sophia Antipolis

S. Icart SMD18 1 / 21

PEVD : eigen vectors/eigen values

An example of application : blind equalization

Some useful definitions on Laurent polynomials

Smith form over the ring of Laurent polynomial matrices

Properties of para-hermitian and para-unitary matrices

Order vs degree of a polynomial matrix

Example of a non diagonalisable PH matrix

and then . . .

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Polynomial Eigenvalue DecompositionLet M(z) be a Laurent polynomial matrix M(z) ∈ Cn×n[z , z−1] :

M(z) =

p∑k=m

Mkzk ,Mk ∈ Cn×n,m, p ∈ Z,m ≤ p

PEVD [McWhirter] :

Given M(z) ∈ Cn×n[z , z−1], find λ(z) ∈ C[z , z−1] , v(z) ∈ Cn[z , z−1] s.t.M(z)v(z) = λ(z)v(z), ∀z

Here, the ”eigenvalues” are (Laurent) polynomial.

Eigenvalue of a polynomial matrix [Lancaster] :roots of det(

∑mk=0 λ

kMk) = 0, so λ ∈ C.

PEVD for para-hermitian matrices and ”orthonormal” eigenvectors(analog : each hermitian matrix can be diagonalized by an unitary matrix)

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Blind EqualizationProblem settingn sources p observations

Channel s1(k)

sn(k)

w1(k)

wp(k)

Equalizer

ŝ1(k)

ŝn(k)

Blind equalization : channel and sources are unknownFind approximation si of sources si

Hypotheses

white i.i.d. sources

number of sources = number of observations

convolutive mixture, FIR channel : C (z) ∈ Cn×n[z−1]

Non unique solution : up to ∆(z)Pwith ∆(z) = diag{z−i} and P permutation

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M(z) s(k) w(k)

Let Γw (z) be the Cross Spectral Density of observations :

Γw (z) = M(z)Γs(z)MH(1

z∗)

if Γs(z) = I , then Γw (z) = M(z)MH( 1z∗ ), and

ΓHw (

1

z∗) = Γw (z)

Γw (z) is a para-hermitian matrix(even if M(z) ∈ Cn×n[z−1] , Γw (z) ∈ Cn×n[z , z−1])

if moreover Γw (z) = I (whitened observations), then

M(z)MH(1

z∗) = I

M(z) is a para-unitary matrix

S. Icart SMD18 5 / 21

Some useful definitions and properties of polynomials

Polynomials Let C[z ] be the ring of polynomials

p(z) =n∑

i=0

pizi with n ∈ N, pi ∈ C

if pn 6= 0, deg(p) = n, moreover if pn = 1, p is monic.if pn = 1 and p0 6= 0, p is L-monic (no roots in 0)(analog definitions for p ∈ C[z−1] )Laurent polynomials Let C[z , z−1] be the ring of Laurent polynomials :

p(z) =n∑

i=m

pizi with m, n ∈ Z,m ≤ n, pi ∈ C

if pmpn 6= 0, d(p) = n −m is the L-degree of p.if p ∈ C[z ], d(p) = deg(p) iff z = 0 is not a root of p.

S. Icart SMD18 6 / 21

C[z , z−1] is an Euclidean ring, so a Principal Ideal Domain (PID)Units of C[z , z−1] : non-zero monomials

p(z) = azα, a ∈ C∗, α ∈ Z

Unicity of gcd : impose the gcd to be L-monic (no roots in 0 neither ∞)

Para-conjugation :

p(z) = p∗(1

z∗), ∀z ∈ C∗

rmq : if p(z) ∈ C[z ] then p(z) ∈ C[z−1].

Para-hermitian polynomial : p = p ⇒ p(z) =d∑

i=−dpiz

i , p−i = p∗i .

Para-unitary polynomial : pp = 1⇒ p(z) = eθzα, θ ∈ R, α ∈ Z.

S. Icart SMD18 7 / 21

Laurent polynomial matrix Cn×n[z , z−1]

a L-polynomial with matrix-valued coefficients

M(z) =

p∑k=m

Mkzk ,Mk ∈ Cn×n,m, p ∈ Z,m ≤ p

or... a matrix with L-polynomial entries

M(z) = (mij(z)) with mij(z) =

p∑k=m

mijkz−k

i and j ”space”-indices and k time-indice

order of M = p −m if Mp and Mm are non zero

S. Icart SMD18 8 / 21

Cn×n[z , z−1] is the ring of L-polynomial matrices

L-unimodular matrices : Units of Cn×n[z , z−1]

det(M(z)) = azα, a ∈ C∗, α ∈ Z

d (det(M(z))) = 0 : zeros at 0 or infinity

Smith form over Cn×n[z , z−1] :Let M ∈ Cn×n[z , z−1], ∃ L-unimodular U1,U2 ∈ Cn×n[z , z−1]

U1(z)M(z)U2(z) = Λ(z)

with Λ(z) diagonal

S. Icart SMD18 9 / 21

Invariant polynomials

Λ(z) =

λ1(z)

. . .

λr (z)

0

0 0

λi unique up to a multiplication by a monomial.

λi divides λi+1.

r normal rank of M .

To ensure unicity : λi L-monic (monic polynomial and no roots in 0)Invariant polynomials can be computed as

λi (z) =∆i (M(z))

∆i−1(M(z))

∆i (M(z)) L-monic gcd of i × i minors of M .S. Icart SMD18 10 / 21

Examples

A(z) =

[z 0z z

], order(A)=0

gcd{z , z , z} = z , det A(z) = z2, so A(z)S∼ zI

L-gcd{z , z , z} = 1, L-gcd{z2} = 1, so A(z)LS∼ I .

B(z) =

[z 0

z − 1 z

], order(B)=1 , B(z)

S∼[

1 00 z2

]and B(z)

LS∼ I

U1(z)B(z)U2(z) = I with

U1(z) =

[1 −1

−z−1 + z−2 z−1

]and U2(z) =

[1 z0 1

](non unique).

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Para-conjugation :

M(z) = MH(1

z∗), ∀z ∈ C∗

then, on the unit circle : M(z) = MH(z).

Para-hermitian property :

M(z) = M(z)

extension of symmetric (M ∈ Rn×n) or hermitian (M ∈ Cn×n) property

Para-unitary property :

M(z)M(z) = I

extension of unitary property

S. Icart SMD18 12 / 21

Some properties of PH and PU matrices

if H(z) is PH H(z) = H(z)

H(z) =d∑

k=−dHkz

k with H−k = HHk ,∀k

the order of H(z) is even (2d)

if U is PU U(z)U(z) = I of order l , U(z) = zm∑l

k=0 Ukzk

l∑k=0

UkUHk = I

l∑j=k

U jUHj−k = 0 ∀k ∈ {1, . . . , l}

S. Icart SMD18 13 / 21

L-invariant polynomials properties

L-unimodular matrix (det(U) = azα) : λi (z) = 1, ∀idet U(z) = azα = det U1(z) det Λ(z) det U2(z),so det Λ(z) = czβ, but λi are L-monic, so λi (z) = 1, ∀i .Para-unitary matrix (UU = I ) : λi (z) = 1, ∀idet U(z)U(z) = 1 = czα

∏ni=1 λi (z)c∗z−α

∏ni=1 λi (z)

so λi invertible and as λi L-monic, λi = 1.

Para-unitary matrixLS∼ L-unimodular matrix

Para-hermitian matrix (H = H) : λi self-inversive

λi (z) = eθi zmi λi (z) with θi ∈ R,mi = deg(λi )

ΛH

LS∼ ΛH but ΛH

is not the L-Smith form of Hbecause ΛH ∈ Cn×n[z−1] (and ΛH = Λ

H∈ Cn×n[z ])

S. Icart SMD18 14 / 21

Order vs Degree

Order defined for polynomial matrices [Kailath, Vaidyanathan],

Degree defined for proper rational matrices : sum of the degrees ofthe denominators of the Smith Mc Millan form, it is the minimumnumber of delays to implement M .

Example :

H(z) =

1z

1z

1z

1+z2

z2

= H−2z−2 + H−1z

−1 + H0, order of H(z) is 2

Let N(z) = z2H(z) polynomial.

Smith form of N : S(z) =

[1 00 z(1 + z2)− z2

]Smith Mc Millan form of H(z) : SM(z) = 1

z2 S(z) =

[ 1z2 0

0 1−z+z2

z

]Mc Millan degree of H(z) is 2+1=3.

S. Icart SMD18 15 / 21

Degree of Laurent polynomial matrices (proposition)

Let H(z) =

p∑k=m

Hkzk ,m ≤ p,Hm and Hp 6= 0.

Define the associated causal matrix (polynomial in z−1) :

H(z) = z−pH(z)

order of H = p −m

L-degree of H : McMillan degree of H .

Example : H(z) =

[1 11 z−1 + z

]= H−1z

−1 + H0 + H1z , order 2

H(z) = z−1H(z) =

[z−1 z−1

z−1 1 + z−2

](previous example), L-degree 3

S. Icart SMD18 16 / 21

Property

If U(z) is para-unitary, its order and its L-degree are equal.

Proof : based on

Vaidyanathan’s factorization of a FIR paraunitary matrix :

U(z) = R0Z (z)R1 . . .Z (z)RN−1Z (z)RN

[1 00 ±1

]

Z (z) =

[1 00 z−1

],R i =

[cos θi sin θi− sin θi cos θi

], N =MacMillan degree.

and that, by definition U0 6= 0

S. Icart SMD18 17 / 21

Diagonalization of a PH matrix

Proposition :

Let H(z) be a PH matrix, does there exist U(z) a PU matrix such that :

U(z)H(z)U(z) = Λ(z)

with Λ(z) a Laurent polynomial diagonal matrix ?

(an hermitian matrix is diagonalizable by a unitary matrix)

not always true (see example)

if exist, ”eigen vectors” are orthonormal w.r.t

〈u(z), v(z)〉 = u(z)v(z) = uH(1

z∗)v(z),∀z ∈ C∗

Λ(z) can be approximated by a rational matrix, and by truncation, apolynomial matrix [Weiss]

S. Icart SMD18 18 / 21

Example

Let H(z) =

[1 11 −2z−1 + 6− 2z

],H = H , det H(z) = −2z−1 + 5− 2z

L-Smith form : S(z) =

[1 00 1− 5

2z + z2

]Suppose ∃ U paraunitary s.t. UHU = Λ ; S LS∼ Λ,So, ”eigen values” have to verify :

L-gcd{λ1, λ2} = 1 and λ1(z)λ2(z) = c ′zα′det H(z).

But λi , if exist, are parahermitian, so (up to a permutation) :λ1(z) = c, c ∈ R∗ and λ2(z) = dzβ(1− 5

2z + z2).

Now, parametrizing H(z)v(z) = cv(z), c ∈ R∗ leads to a system withoutany solution.There exists no polynomial unitary matrix s.t. UHU = Λ.(H as order 2 but degree 4)

S. Icart SMD18 19 / 21

The PEVD has not always an exact solution.

There exits approximated solution at least on the unit circle.

There exit iterative algorithms (Jacobi type, Extended Givens rotationwith delay).

Open problem

On which condition does there exist a polynomial solution ? (degreevs order ) ?

Can we write this problem with tensors of order 3 (coefficientrelations of PH and PU matrices) ?

S. Icart SMD18 20 / 21

Bibliography :

J. G. McWhirter et al. An EVD algorithm for Para-Hermitian polynomialmatrices, IEEE T. Signal Processing, 2007

T. Kailath : Linear Systems, Prentice Hall, Information and SystemSciences Series, 1980

P.P. Vaidyanathan. Multirate Systems and Filter Bank, Prentice Hall, SignalProcessing Series, 1993

S. Weiss et al. On the existence and uniqueness of the eigenvaluedecomposition of a parahermitian matrix, T. Signal Processing, 2018

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