Majorization inequalities for valuations of eigenvalues Marianne - - PowerPoint PPT Presentation

Majorization inequalities for valuations of eigenvalues

Marianne Akian INRIA Saclay - ˆ Ile-de-France and CMAP, ´ Ecole polytechnique CNRS SYMBIONT ANR/DFG Kick-off Meeting Bonn, Jul 4-5, 2018 Based on works with Ravindra Bapat, St´ ephane Gaubert, Andrea Marchesini, Meisam Sharify, and Fran¸ coise Tisseur

Majorizations of scalar polynomial roots

Let ζ1, . . . , ζd, |ζ1| ≥ · · · ≥ |ζd|, denote the roots of the polynomial P = a0 + a1z + · · · + akzk + · · · + adzd, ai ∈ C . Let α1 ≥ · · · ≥ αd denote the inclinaisons num´ eriques of P, that is the exponential

• f the opposites of the slopes of the New-

ton polygon of P, concave(k → log |ak|).

2 4 6 8 10 12 14 16 2 4 6 8 10

For each k ≥ 1, there exists Lk, Uk > 0 independent of d such that Lk α1 · · · αk ≤ |ζ1 · · · ζk| ≤ Uk α1 · · · αk , with L−1

k

= d

k

• and Uk ≤
• (k + 1)k+1/kk.

(Ostrowski, “Recherches sur la m´ ethode de Graeffe et les z´ eros des polynomes et des s´ eries de Laurent”, Acta Math 72, 99-257, 1940)

Majorizations of scalar polynomial roots

Let ζ1, . . . , ζd, |ζ1| ≥ · · · ≥ |ζd|, denote the roots of the polynomial P = a0 + a1z + · · · + akzk + · · · + adzd, ai ∈ C . Let α1 ≥ · · · ≥ αd denote the inclinaisons num´ eriques of P, that is the exponential

• f the opposites of the slopes of the New-

ton polygon of P, concave(k → log |ak|).

2 4 6 8 10 12 14 16 2 4 6 8 10

For each k ≥ 1, there exists Lk, Uk > 0 independent of d such that Lk α1 · · · αk ≤ |ζ1 · · · ζk| ≤ Uk α1 · · · αk , with L−1

k

= d

k

• and Uk ≤
• (k + 1)k+1/kk.

Ostrowski proved Lk (which is optimal) and reproduced the proof of Uk given by P´

• lya

Majorizations of scalar polynomial roots

Let ζ1, . . . , ζd, |ζ1| ≥ · · · ≥ |ζd|, denote the roots of the polynomial P = a0 + a1z + · · · + akzk + · · · + adzd, ai ∈ C . Let α1 ≥ · · · ≥ αd denote the inclinaisons num´ eriques of P, that is the exponential

• f the opposites of the slopes of the New-

ton polygon of P, concave(k → log |ak|).

2 4 6 8 10 12 14 16 2 4 6 8 10

For each k ≥ 1, there exists Lk, Uk > 0 independent of d such that Lk α1 · · · αk ≤ |ζ1 · · · ζk| ≤ Uk α1 · · · αk , with L−1

k

= d

k

• and Uk ≤
• (k + 1)k+1/kk.

(Ostrowski, P´

• lya, 1940)

Previous inequalities include: Uk ≤ k + 1 (Hadamard, 1891), |ζ1| ≤ 2α1 (Fujiwara, 1916), |ζ1 · · · ζk| ≤ (k + 1)αk

1 (Specht, 1938), Uk ≤ 2k + 1 in

(Ostrowski, 1940).

Max-plus or tropical algebra

Let T = Rmax := (R ∪ {−∞}, ⊕, ⊗) be the additive tropical or max-plus idempotent semifield, where a ⊕ b = max(a, b) and a ⊗ b = a + b, with neutral elements 0 = −∞ and 1 = 0. On can define vectors, matrices, polynomials, eigenvalues, eigenvectors, permanents,... T is the limit of the logarithmic deformation of R+ semiring: max(a, b) = limε→0+

log(ε−a+ε−b) − log(ε)

a + b =

log(ε−aε−b) − log(ε)

. And it gives bounds: max(a, b) ≤ ε log(ea/ε + eb/ε) ≤ ε log(2) + max(a, b) . It is isomorphic to the multiplicative tropical semifield Tm := (R≥0, max, ×), by x → exp(x).

Valuations

Let K be a field, a map v : K → R ∪ {−∞} is a non- archimedean valuation if v(s1 + s2) ≤ max(v(s1), v(s2)), v(s1s2) = v(s1) + v(s2) v(s) = −∞ ⇔ s = 0 . Then v(s1 + s2) = max(v(s1), v(s2)) if v(s1) = v(s2). Ex: K = C{{ǫ}}, the field of (general- ized) Puiseux series, v(s) = − val(s), e.g. v(ǫ−1/3 + 3 − 8ǫ2 + · · · ) = 1/3. a map v : K → R+ is an archimedean valuation if v(z1 + z2) ≤ v(z1) + v(z2), v(z1z2) = v(z1) v(z2), v(z) = 0 ⇔ z = 0 . Then v(z1 + z2) ≤ 2 max(v(z1), v(z2)). Ex: K = C, v(z) = |z|. Given a (archimedean or non-archimedean) valuation v on K, one apply v entrywise on K n, K n×n, K[X], e.g. v(z1, . . . , zn) := (v(z1), . . . , v(zn)). Given a variety V on K, its image by v is called the amoeba of V .

Let Tm = (R≥0, max, ×) be the multiplicative tropical semifield

The “inclinaisons num´ eriques” αi of P = a0 + a1z + · · · + akzk + · · · + adzd are the tropical roots (nondifferentiability locus) of the multiplicative tropical polynomial v(P) ∈ Tm[X]: v(P)(x) = max

0≤j≤d(|aj|xj)

⇔ the log αi are the tropical roots of p(x) = max0≤j≤d(log |aj| + jx). Here v is the archimedean valuation : v : C → R≥0, x → |x|.

Let Tm = (R≥0, max, ×) be the multiplicative tropical semifield

The “inclinaisons num´ eriques” αi of P = a0 + a1z + · · · + akzk + · · · + adzd are the tropical roots (nondifferentiability locus) of the multiplicative tropical polynomial v(P) ∈ Tm[X]: v(P)(x) = max

0≤j≤d(|aj|xj)

⇔ the log αi are the tropical roots of p(x) = max0≤j≤d(log |aj| + jx). Since the Newton polygon of P is the graph of the convave hull of the map j → log |aj| and p is its Legendre-Fenchel transform.

2 4 6 8 10 12 14 16 2 4 6 8 10 −6 −5 −4 −3 −2 −1 1 2 5 10 15 20 25 30 35

Newton polygon of P Graph of p Example: P = 1 + eX + e6X2 + e4X4 + e9X8 + e5X10 + eX16. Tropical roots: −3, −1/2, and 1, with multiplicities 2, 6 and 8 resp.

Theorem ( See Kapranov, 2006) Let K be an algebraically closed field with a (non-archimedean) valuation v, and f =

k fkX k ∈ K[X], then the valuations of the roots of f (counted with

multiplicities) coincide with the tropical roots of the tropical polynomial v(f ) := n

k=0 v(fk)X k.

Follows from Puiseux theorem in dimension 1, and is derived from Puiseux in other dimensions.

Majorizations of scalar polynomial roots

Let ζ1, . . . , ζd, |ζ1| ≥ · · · ≥ |ζd|, denote the roots of the polynomial P = a0 + a1z + · · · + akzk + · · · + adzd, ai ∈ C . Let α1 ≥ · · · ≥ αd denote the inclinaisons num´ eriques of P, that is the exponential

• f the opposites of the slopes of the New-

ton polygon of P, concave(k → log |ak|).

2 4 6 8 10 12 14 16 2 4 6 8 10

For each k ≥ 1, there exists Lk, Uk > 0 independent of d such that Lk α1 · · · αk ≤ |ζ1 · · · ζk| ≤ Uk α1 · · · αk , with L−1

k

= d

k

• and Uk ≤
• (k + 1)k+1/kk.

(Ostrowski, P´

• lya, 1940)

Let Log : (C∗)d → Rd, x → (log |x1|, . . . , log |xd|). Ostrowski bounds give the distance between the archimedan amoeba A = Log(V )

• f the finite set V of Cd of solutions (ζ1, . . . , ζd) of
• i1<···<ik

ζi1 · · · ζik = (−1)kad−k/ad, k = 1, . . . d , up to permutations of coordinates, to its “tropical skeleton” T .

Let Log : (C∗)d → Rd, x → (log |x1|, . . . , log |xd|). Ostrowski bounds give the distance between the archimedan amoeba A = Log(V )

• f the finite set V of Cd of solutions (ζ1, . . . , ζd) of
• i1<···<ik

ζi1 · · · ζik = (−1)kad−k/ad, k = 1, . . . d , up to permutations of coordinates, to its “tropical skeleton” T . T is the set of (log α1, . . . , log αd) solutions of |ad−k/ad| ⊕

• i1<···<ik

αi1 · · · αik “ = ” 0, k = 1, . . . d . Ostrowski bounds implies d(A, T ) ≤ 1 for the distance d(x, y) = minσ,σ′∈Σd (xσ(i)) − (yσ′(i)) associated to the weak Minkowski (non symmetric) norm: x = max

k=1,...,d

• (log Uk)−1(x1 + · · · + xk)+ − (log Lk)−1(x1 + · · · + xk)−

.

These are different from the bounds on the distance between the archimedan amoeba A = Log(V ) of the finite set V of roots of P and its tropical skeleton:

• such bounds can be obtained by using Rouch´

e theorem.

• they are related to lopsided approximations.
• see in particular the works of (Gaubert, Sharify, 2011), (Bini, Noferini, Sharify,

2013), (Erg¨ ur, Paouris, Rojas, 2014).

In the multivariate case (non zero dimension), such bounds appeared in:

(Passare, R¨ ullgaard, 04); (Purbhoo, 06); (Avendano, Kogan, Nisse, Rojas, 13), (Forsg˚ ard, 16), (Forsg˚ ard, Matusevich, Mehlhop, de Wolff, 17).

This work: give majorizations/Ostrowski bounds for the eigen- values of matrix polynomials

This work: give majorizations/Ostrowski bounds for the eigen- values of matrix polynomials

We are interested in

• Non-archimedean case
• Archimedean case
• Eigenvectors
• Condition numbers and applications to numerical computations of eigenvalues

and eigenvectors

This work: give majorizations/Ostrowski bounds for the eigen- values of matrix polynomials

We are interested in

• Non-archimedean case
• Archimedean case
• Eigenvectors
• Condition numbers and applications to numerical computations of eigenvalues

and eigenvectors

Eigenvalues of tropical matrices

Let T be the max-plus semifield: T = Rmax := (R ∪ {−∞}, max, +) with its laws denoted ⊕ and ⊗ (⊗ may be omited).

• For a matrix A = (Aij)i,j=1,...,n ∈ Tn×n, we define

tper A := ⊕σ∈Σn ⊗n

i=1 Aiσ(i) .

• this is the optimal assignment value associated to A:

tper A = max

σ∈Σn n

• i=1

Aiσ(i) .

Eigenvalues of tropical matrices

Let T be the max-plus semifield: T = Rmax := (R ∪ {−∞}, max, +) with its laws denoted ⊕ and ⊗ (⊗ may be omited).

• For a matrix A = (Aij)i,j=1,...,n ∈ Tn×n, we define

tper A := ⊕σ∈Σn ⊗n

i=1 Aiσ(i) .

• this is the optimal assignment value associated to A:

tper A = max

σ∈Σn n

• i=1

Aiσ(i) .

• The characteristic polynomial function of A is the tropical polynomial function

PA(x) = tper(A ⊕ xI) (parametric optimal assignment) and its tropical algebraic eigenvalues are the tropical roots of PA.

Eigenvalues of tropical matrices

Let T be the max-plus semifield: T = Rmax := (R ∪ {−∞}, max, +) with its laws denoted ⊕ and ⊗ (⊗ may be omited).

• For a matrix A = (Aij)i,j=1,...,n ∈ Tn×n, we define

tper A := ⊕σ∈Σn ⊗n

i=1 Aiσ(i) .

• this is the optimal assignment value associated to A:

tper A = max

σ∈Σn n

• i=1

Aiσ(i) .

• The characteristic polynomial function of A is the tropical polynomial function

PA(x) = tper(A ⊕ xI) (parametric optimal assignment) and its tropical algebraic eigenvalues are the tropical roots of PA.

They are not related to (⊂ and ⊃) the scalars x ∈ T such that A ⊕ xI is

tropically singular, meaning “tper(A ⊕ xI) = 0”.

Eigenvalues of tropical matrices

Let T be the max-plus semifield: T = Rmax := (R ∪ {−∞}, max, +) with its laws denoted ⊕ and ⊗ (⊗ may be omited).

• For a matrix A = (Aij)i,j=1,...,n ∈ Tn×n, we define

tper A := ⊕σ∈Σn ⊗n

i=1 Aiσ(i) .

• this is the optimal assignment value associated to A:

tper A = max

σ∈Σn n

• i=1

Aiσ(i) .

• The characteristic polynomial function of A is the tropical polynomial function

PA(x) = tper(A ⊕ xI) (parametric optimal assignment) and its tropical algebraic eigenvalues are the tropical roots of PA.

They are different from the geometric tropical eigenvalues of A, that is

solutions λ of Ax = λx for some x ∈ Tn \ {0}.

Eigenvalues of tropical matrices

• The characteristic polynomial function of a n × n matrix and its roots can be

computed using at most n optimal assignment problems (Burkard and

Butkovic, 2003) or directly by using a parametric flow algorithm in O(n3) (Gassner and Klinz, 2010).

• The formal characteristic polynomial of A is defined as

PA = tper(A ⊕ XI) = ⊕n

k=0 trmax k

(A)X n−k , where trmax

k

(A) =

• J, |J|=k

tper AJ,J .

• It is not known whether PA (or the trmax

k

(A)) can be computed in polynomial time.

Non-archimedean valuations of eigenvalues

Let K be an algebraically closed field with a (non-archimedean) valuation v. Let A = (Aij) ∈ K n×n, and let A = (Aij) with Aij = v(Aij).

• 1. By Kapranov theorem, the valuations of the eigenvalues λ1, . . . , λn of A

(counted with multiplicities) coincide with the tropical roots of

n

• k=0

v(trk(A))X n−k . By 1: If v(trk(A)) = trmax

k

(A) for all k, the valuations of the eigenvalues coincide with the tropical algebraic eigenvalues of A. But how can we check that v(trk(A)) = trmax

k

(A) for all k? (we do not even know whether computing trmax

k

(A) is in P!). What can we say if v(trk(A)) = trmax

k

(A) ?

Weak-majorization of non-archimedean valuations of eigenval- ues

Theorem ( A., Bapat, Gaubert, 2004 and 2016) Let K be an algebraically closed field with a (non-archimedean) valuation v. Let A = (Aij) ∈ K n×n. Then, the sequence of valuations of the eigenvalues λ1, . . . , λn of A (counted with multiplicities) is weakly majorized by the sequence

• f tropical (algebraic) eigenvalues α1, . . . , αn of the matrix

A = v(A) := (v(Aij)) ∈ Tn×n, that is v(λ1) + · · · + v(λk) ≤ α1 + · · · + αk, k = 1, . . . , n , if v(λ1) ≥ · · · ≥ v(λk) and α1 ≥ · · · ≥ αn.

Matrix polynomials

• A matrix polynomial over the field (or ring) K is a formal polynomial with

coefficients in K n×n for some n: A = A0 + XA1 + · · · + X dAd, Ai ∈ K n×n ,

• r equivalently a n × n matrix with entries in K[X].
• It is said regular if the polynomial det(A(x)) is not identically zero on K.
• Then its finite eigenvalues are the roots of the polynomial det(A(x)).
• If the degree D of det(A(x)) is < nd, then ∞ is an eigenvalue of multiplicity

nd − D.

Tropical matrix polynomials

• A tropical matrix polynomial is a formal polynomial with coefficients in Tn×n

for some n: A = A0 ⊕ XA1 ⊕ · · · ⊕ X dAd, Ai ∈ Tn×n ,

• r equivalently a n × n matrix with entries in T[X].
• It is said regular if the tropical polynomial function tper(A(x)) is not

identically zero on T.

• Then its finite tropical algebraic eigenvalues are the tropical roots of the

tropical polynomial function tper(A(x)).

• If the degree D of tper(A(x)) is < nd, then +∞ is a tropical eigenvalue of

multiplicity nd − D.

Non-archimedean valuations of eigenvalues: the generic case

Let K = C{{ǫ}}, and v its valuation. Let A be a regular matrix polynomial: A = A0 + XA1 + · · · + X dAd, Ai ∈ K n×n . Consider the tropical matrix polynomial: A = v(A) := A0 ⊕ · · · ⊕ X dAd, with Ak = v(Ak), k = 0, . . . , d . and assume that (Ak)ij ∼ (ak)ijǫ−(Ak)ij. Theorem ( A., Bapat, Gaubert, 2004 and 2016) Then, A is a regular tropical matrix polynomial. Moreover, generically in the dominant coefficients (ak)ij of the (Ak)ij, the valuations of the eigenvalues of A counted with their multiplicities are the tropical algebraic eigenvalues of the tropical matrix polynomial A. But, how to check genericity?

Hungarian pairs

• For a matrix A ∈ Tn×n, tper A is the optimal assignment value:

tper A = max

σ∈Σn n

• i=1

Aiσ(i) .

• This can be cast as a LP over the set of bi-stochastic matrices.
• The dual LP is

tper A = min

U,V ∈Rn

  

n

• i=1

Ui +

n

• j=1

Vj, Aij ≤ Ui + Vj ∀i, j = 1, . . . , n    .

• We call Hungarian pair with respect to A an optimal dual solution (U, V ) as

it is produced by Kuhn’s Hungarian algorithm (which actually goes back to Jacobi).

• We call saturation graph:

Sat(A, U, V ) := {(i, j) ∈ {1, . . . , n}2 | Aij = Ui + Vj} .

• Given an arbitrary Hungarian pair (U, V ), a permutation σ is optimal if and
• nly if (i, σ(i)) ∈ Sat(A, U, V ) for all i.

First order asymptotics of eigenvalues

Let K = C{{ǫ}}, A = A0 + XA1 + · · · + X dAd, Ak ∈ K n×n, A = v(A) := A0 ⊕ · · · ⊕ X dAd, Ak = v(Ak) and (Ak)ij ∼ (ak)ijǫ−(Ak)ij. Theorem ( A., Bapat, Gaubert, 2004 and 2016) Let γ ∈ R denote any finite algebraic eigenvalue of A, and mγ,A its multiplicity. Let (U, V ) be a Hungarian pair with respect to A(γ) and G = Sat(A(γ), U, V ). Consider the matrix polynomial over C: a(γ) := a(γ) + Xa(γ)

1

+ · · · + X da(γ)

d

, (a(γ)

k )ij =

• (ak)ij

if (i, j) is an arc of G and γk(Ak)ij = Aij(γ) = 0,

• therwise.

Suppose a(γ) is regular and let λ1, . . . , λmγ be the non-zero eigenvalues of a(γ). Then mγ ≤ mγ,A and the matrix polynomial Aǫ has mγ eigenvalues Lǫ,1, . . . , Lǫ,mγ with first order asymptotics of the form Lǫ,i ∼ λiǫ−γ. Moreover, generically in the (ak)ij, we have the equality: mγ = mγ,A.

The genericity condition holds if the “extremal degree coefficients” of the complex matrix polynomial a(γ) are nonzero. We obtain certificates of genericity. The proof of the theorem is based on: Lemma Consider the diagonal scaling Bǫ = diag(ǫU)Aǫ(ǫ−γX) diag(ǫV ) . Then, lim

ǫ→0 Bǫ = a(γ) .

Proof. Let A be a tropical matrix and let (U, V ) be a Hungarian pair with respect to A, then the matrix B = diag(−U)A diag(−V ) satisfies Bij ≤ 0, i, j = 1, . . . , n, with equality on the saturation graph G = Sat(A, U, V ). Related ideas in an algorithm of

• K. Murota, 1990.

Summary of following results

• Application to singular cases of Lidskii theorem A., Bapat, Gaubert, 2004.
• Asymptotics of eigenvectors A., Gaubert, Marchesini, Tisseur, 2015.
• Asymptotics and bounds of condition number of eigenvalues A., Gaubert,

Marchesini, Tisseur, 2015.

• Application of the former diagonal scaling to improve the condition number

and so the numerical computation of eigenvalues A., Gaubert, Marchesini,

Tisseur, 2015.

• Majorization of archimedean valuations of eigenvalues (Ostrowski-P´
• lya type

bounds) A., Gaubert, Marchesini, LAA 2014.

• Matrix polynomial P´
• lya inequality A., Gaubert, Sharify, LAA 2016.

A generalization of Friedland inequality

Let λ1, . . . , λn, |λ1| ≥ · · · ≥ |λn|, denote the eigenvalues of A ∈ Cn×n (counted with multiplicities). Let γ1 ≥ · · · ≥ γn denote the algebraic tropical (multiplicative) eigenvalues of v(A) = (|Aij|)ij. Denote: pat A ∈ Rn×n, (pat A)ij = 1 if Aij = 0 and 0 otherwise. Theorem ( A., Gaubert, Marchesini, LAA 2014) |λ1| · · · |λk| ≤ Uk(pat A)γ1 · · · γk, k = 1, . . . , n , with Uk(B) = ρ( k

per(B)) ≤

n k

• k! ,

for B ∈ {0, 1}n×n, 1 ≤ k ≤ n . k

per(B) ∈ C(

n k)×( n k) is indexed by the subsets of [n] of cardinal k and such that:

k

perB

• IJ

= per BIJ , where per is the usual permanent.

Numerical approximation of eigenvalues

We compute the eigenvalues of a (randomly generated) matrix polynomial Aǫ with degree d = 4 and size n = 5, for several values of ǫ → 0, using

• MATLAB’s polyeig function (which applies a companion linearization

followed by QZ algorithm).

• Arbitrary precision computation

0 ← ǫ

10 -15 10 -10 10 -5 10 0

Magnitude of eigenvalues

10 -20 10 -10 10 0 10 10 10 20

arbitrary precision polyeig

Numerical approximation of eigenvalues

We compute the eigenvalues of a (randomly generated) matrix polynomial Aǫ with degree d = 4 and size n = 5, for several values of ǫ → 0, using

• Arbitrary precision computation
• polyeig after applying the Hungarian diagonal scaling

Bǫ = diag(ǫU)Aǫ(ǫ−γX) diag(ǫV ) for each tropical eigenvalue γ of A: one selects the mγ eigenvalues of Bǫ the modulus of which are the closest to 1 and then do the inverse scaling on these eigenvalues.

0 ← ǫ

10 -15 10 -10 10 -5 10 0

Magnitude of eigenvalues

10 -20 10 -10 10 0 10 10 10 20

arbitrary precision Hungarian scaling

0 ← ǫ

10 -15 10 -10 10 -5 10 0

Condition number

10 -20 10 0 10 20 10 40 10 60 10 80

before scaling, using polyeig eigenvalues before scaling, using "true" eigenvalues after Hungarian scaling

Conclusion

• In the non-archimedean case, the valuations of the eigenvalues of matrix

polynomials are equal to their tropical analogues in generic situations. We have the same for eigenvectors, and eigenvalue condition numbers.

• Some of these equalities can be replaced by inequalities without any condition.
• In the archimedean case, similar unconditional inequalities have been shown

for eigenvalues and eigenvalue condition numbers.

• The Hungarian diagonal scaling is the main theoretical tool and it improves in

general the condition number and so the numerical computation of eigenvalues.

• What about amoebas of general (dim 0) varieties? Unconditional inequalities,

scalings?

• M. Akian, R. Bapat, and S. Gaubert. Perturbation of eigenvalues of matrix pencils and the
• ptimal assignment problem. C. R. Math. Acad. Sci. Paris, 339(2):103–108, 2004.
• M. Akian, R. Bapat, and S. Gaubert. Non-archimedean valuations of eigenvalues of matrix
• polynomials. Linear Algebra Appl., 498:592–627, 2016. See also arXiv:1601.00438.
• M. Akian, S. Gaubert, and A. Marchesini. Tropical bounds for eigenvalues of matrices. Linear

Algebra Appl., 446:281–303, 2014. See also arXiv:1309.7319.

• M. Akian, S. Gaubert, A. Marchesini, and F. Tisseur. Tropical diagonal scaling of matrix
• polynomials. In preparation, 2015.
• A. Marchesini. Tropical methods for the localization of eigenvalues and application to their

numerical computation. PhD thesis, Ecole Polytechnique, France, 2015. See pastel.archives-ouvertes.fr/tel-01285110v1.

• M. Akian, S. Gaubert, and M. Sharify. Log-majorization of the moduli of the eigenvalues of a

matrix polynomial by tropical roots, 2013. Linear Algebra Appl., Published online, 2016. See also arXiv:1304.2967.

• K. Murota. Computing Puiseux-series solutions to determinantal equations via combinatorial
• relaxation. SIAM J. Comput., 19(6):1132–1161, 1990.
• T. Betcke. Optimal scaling of generalized and polynomial eigenvalue problems. SIAM J. Matrix Anal.

Appl., 30(4):1320–1338, 2008/09.

• H.-Y. Fan, W-W. Lin, and P. Van Dooren. Normwise scaling of second order polynomial
• matrices. SIAM J. Matrix Anal. Appl., 26(1):252–256, 2004.
• S. Gaubert and M. Sharify. Tropical scaling of polynomial matrices. In Rafael Bru and Sergio

Romero-Viv´

• , editors, Proceedings of the third Multidisciplinary International Symposium on Positive

Systems: Theory and Applications (POSTA 09), volume 389 of LNCIS, pages 291–303, Valencia, Spain,

• 2009. Springer.
• A. Melman. Generalization and variations of Pellet’s theorem for matrix polynomials. Linear Algebra

Appl., 439(5):1550–1567, 2013.

• D.A. Bini, V. Noferini, and M. Sharify. Locating the Eigenvalues of Matrix Polynomials. SIAM
• J. Matrix Anal. Appl., 34(4):1708–1727, 2013.
• V. Noferini, M. Sharify, and F. Tisseur. Tropical roots as approximations to eigenvalues of

matrix polynomials. SIAM J. Matrix Anal. Appl., 36(1):138–157, 2015.

Slides continued

An example

Consider Aǫ = Aǫ,0 − XI, and Aǫ,0 =    b11 b12 b13 b21 b22ǫ b23ǫ b31 b32ǫ b33ǫ    , where bij ∈ C. which is singular for Lidskii theorem.

• The associated tropical matrix polynomial and characteristic polynomial are:

A =    0 ⊕ X (−1) ⊕ X −1 −1 (−1) ⊕ X    , PA(x) = (x ⊕ 0)2(x ⊕ (−1)) . with eigenvalues γ1 = γ2 = 0, and γ3 = −1.

• For γ = 0, U = V = (0, 0, 0) yields a Hungarian pair with respect to

A(0) =    001 00 00 00 01 −1 00 −1 01    , where the subscript k of the entry Aij(γ) means that (i, j) ∈ Sat(A(γ), U, V ) and that γk(Ak)ij = Aij(γ) = 0.

• The eigenvalues of a(0) are the roots of

det    b11 − λ b12 b13 b21 −λ b31 −λ    = λ(−λ2 + λb11 + b12b21 + b31b31) = 0 .

• The certificate of genericity is b12b21 + b31b31) = 0.
• This equation has, for generic values of the parameters bij, two non-zero roots,

λ1, λ2, which yields two eigenvalues of Aǫ, Lǫ,m ∼ λmǫ0 = λm, for m = 1, 2.

• For γ = −1, we can take U = (0, 1, 1), V = (−1, 0, 0), so that

A(1) =    00 00 00 (−1)01 (−1)0 00 (−1)0 (−1)01    , det    b12 b13 b21 b22 − λ b23 b31 b31 b33 − λ    = 0 .

• The latest equation yields

λ(b12b21 + b13b31) + b12b23b31 + b13b32b21 − b21b12b33 − b31b13b22 = 0.

• This equation has generically a unique nonzero root, λ1, and there is a branch

Lǫ,1 ∼ λ1ǫ.

This work: give majorizations/Ostrowski bounds for the eigen- values of matrix polynomials

• Non-archimedean case
• Archimedean case
• Eigenvectors
• Condition numbers and applications to numerical computations of eigenvalues

and eigenvectors

Majorizations of matrix eigenvalues

Let λ1, . . . , λn, |λ1| ≥ · · · ≥ |λn|, denote the eigenvalues of A ∈ Cn×n (counted with multiplicities). Let γ1 ≥ · · · ≥ γn denote the algebraic tropical (multiplicative) eigenvalues of v(A) = (|Aij|)ij. Denote: pat A ∈ Rn×n, (pat A)ij = 1 if Aij = 0 and 0 otherwise. Theorem ( Friedland, 1988) If A ∈ Rn×n

+

, then ρ(A) ≤ ρ(pat A)ρ∞(A) ≤ nρ∞(A), where ρ∞(A) = max

1≤k≤n max i1,...,ik(Ai1i2 · · · Aiki1)

1 k .

Corollary For A ∈ Cn×n: |λ1| ≤ ρ(pat A)γ1 ≤ nγ1. Uses ρ(A) ≤ ρ(v(A)) and ρ∞(A) = ρmax(A) = γ1.

A generalization of Friedland inequality

Theorem ( A., Gaubert, Marchesini, LAA 2014) |λ1| · · · |λk| ≤ Uk(pat A)γ1 · · · γk, k = 1, . . . , n , with Uk(B) = ρ( k

per(B)) ≤

n k

• k! ,

for B ∈ {0, 1}n×n and 1 ≤ k ≤ n. Here, for all B ∈ Cn×n and 1 ≤ k ≤ n, k

per(B) and k(B) ∈ C(

n k)×( n k), are

indexed by the subsets of [n] of cardinal k and are such that: k B

• IJ

= det BIJ k

perB

• IJ

= per BIJ , where per is the usual permanent.

Ingredients of the proof

Theorem (Log-convexity of the spectral radius

(Kingman, 61))

Let A and B ∈ Rn×n

+

, and 0 ≤ α ≤ 1, then ρ(A[α] ◦ B[1−α]) ≤ ρ(A)αρ(B)1−α . where A[r] is the entrywise r-th power of A and ◦ denotes the Hadamard (entrywise) product of matrices. Corollary For A, B ∈ Rn×n

+

, we have ρ(A ◦ B) ≤ ρ(A)ρmax(B). Theorem ( A., Gaubert, Marchesini, LAA 2014) Let A ∈ Tn×n, and let γ1 ≥ · · · ≥ γn be its tropical eigenvalues. Define k

max(A)

using tropical (multiplicative) permanents. Then, ρmax( k

maxA) ≤ γ1 · · · γk,

k = 1, . . . , n .

• This bound is not tight for companion matrices of polynomials:

Uk(pat A) ≤ min(k + 1, n − k + 1), so we recover the bound of roots of polynomials by Hadamard but not the one of P´

• lya.
• This bound is not tight for full matrices A ∈ Cn×n: if k = n, then

Uk(pat A) = n!, whereas, the following improvement shows |λ1 · · · λn| ≤ nn/2γ1 · · · γn .

An improvement of the majorization of eigenvalues

Theorem |λ1 . . . λk| ≤ Ukγ1 . . . γk, k = 1, . . . , n , with Uk = n k

• kk/2 .

In particular, | det(A)| ≤ nn/2 tper(v(A)) . Proof. Let (U, V ) be a Hungarian pair with respect to log(v(A) ⊕ γkI). Apply the diagonal scaling B = diag(exp(−U)A diag(exp(−V )) . Then v(B) ≤ pat A, k(A) = k(diag(exp(U))k(B)k(diag(exp(V )), and v(k(A)) ≤ γ1 . . . γk v(k(B)). By Hadamard inequality, |k(B)IJ| ≤ kk/2.

• The proof shows that we can replace Uk by: U′

k(pat A) with

U′

k(B) = max

• ρ(|

k (C)|), C ∈ Cn×n, v(C)ij ≤ Bij∀i, j = 1, . . . , n

• .
• We have

U′

k(B) ≤ Uk(B) := ρ(

k

mod(B)) ≤ Uk(B) ,

where k

mod(B) ∈ R(

n k)×( n k)

+

is such that: k

modB

• IJ

= max

• | det C|, C ∈ Ck×k, v(C) ≤ BIJ
• .
• For full matrices, Uk =

n

k

• kk/2, by Hadamard inequality.
• For k = n, and full matrices, U′

n = Un = nn/2, and the bound is tight.

• If 1 < k < n, the new bound is still not tight.
• For companion matrices, the bound is the same as before:

Uk(pat(A)) = Uk(pat(A)).

Remarks

• The above upper bound is unconditional.
• It can be applied to matrix polynomials A = A0 + XA1 + · · · + X dAd,

Aj ∈ Cn×n, 0 ≤ j ≤ d such that Ad = I.

• A lower bound was obtained ( A., Gaubert, Marchesini, LAA 2014) under

technical conditions only since in the non-archimedean case the equality holds

• nly generically.

This work: give majorizations/Ostrowski bounds for the eigen- values of matrix polynomials

• Non-archimedean case
• Archimedean case
• Eigenvectors
• Condition numbers

Non-archimedean valuations and asymptotics of eigenvectors

Let K = C{{ǫ}}, A be a regular matrix polynomial: A = A0 + XA1 + · · · + X dAd, Ak ∈ K n×n, A = v(A) := A0 ⊕ · · · ⊕ X dAd, Ak = v(Ak) and (Ak)ij ∼ (ak)ijǫ−(Ak)ij. Let γ ∈ R denote any finite algebraic eigenvalue of A, and mγ,A its multiplicity. Let (U, V ) be a Hungarian pair with respect to A(γ) and G = Sat(A(γ), U, V ). Consider the matrix polynomial over C: a(γ) := a(γ) + Xa(γ)

1

+ · · · + X da(γ)

d

, (a(γ)

k )ij =

• (ak)ij

if (i, j) is an arc of G and γk(Ak)ij = Aij(γ) = 0,

• therwise.

Suppose a(γ) is regular. Consider the diagonal scaling Bǫ = diag(ǫU)Aǫ(ǫ−γX) diag(ǫV ) . Then, lim

ǫ→0 Bǫ = a(γ) .

Non-archimedean valuations and asymptotics of eigenvectors

Theorem ( A., Gaubert, Marchesini, Tisseur, 2015) Assume that λ is a simple non-zero eigenvalue of a(γ), and let z (resp. w) be a right (resp. left) eigenvector of a(γ) for the eigenvalue λ. Then,

• For ǫ > 0 small enough, Aǫ has a simple non-zero eigenvalue Lǫ ∼ λǫ−γ.
• One can choose a right (resp. left) eigenvector Zǫ (resp. Wǫ) of Aǫ w.r.t. Lǫ

so that zi = lim

ǫ→0 ǫ−Vi(Zǫ)i,

wi = lim

ǫ→0(Wǫ)iǫ−Ui,

∀i ∈ [n] . In particular v(Z) ≤ −V , v(W) ≤ (−U)T with equality of one entry at least.

Non-archimedean valuations and asymptotics of eigenvectors

Theorem ( A., Gaubert, Marchesini,Tisseur, 2015)

• For generic values of the parameters (ak)ij, the matrix polynomial a(γ) is

regular, it has mγ,A non-zero eigenvalues, and all these non-zero eigenvalues are simple.

• Assume that G = Sat(A(γ), U, V ) is fully indecomposable.

Which means that the adjacency matrix of G cannot be transformed into a block triangular matrix by applying (different) permutations of rows and columns. Then, for generic values of the parameters (ak)ij, for each nonzero simple eigenvalue λ, the entries of the associated right and left eigenvectors of a(γ), z and w, are nonzero.

• In that case v(Z) = −V ,

v(W) = (−U)T.

• −V plays the role of a right eigenvector of A associated to γ;
• (−U)T plays the role of a left eigenvector of A associated to γ.

This work: give majorizations/Ostrowski bounds for the eigen- values of matrix polynomials

• Non-archimedean case
• Archimedean case
• Eigenvectors
• Condition numbers

Asymptotics of condition numbers

The normwise relative condition number of the eigenvalue L of A is κ(A, L) = lim sup

ǫ→0

|∆L| ǫ |L|

• L + ∆L is an eigenvalue of A + ∆A,

∆Ak2 ≤ ǫ Ak2 for k = 0, . . . , d

• .

It satisfies (see Tisseur, 2000): κ(A, L) = W2Z2(d

k=0 Ak|L|k)

|L||WA′(L)Z| .

Asymptotics of condition numbers

The normwise relative condition number of the eigenvalue L of A is κ(A, L) = lim sup

ǫ→0

|∆L| ǫ |L|

• L + ∆L is an eigenvalue of A + ∆A,

∆Ak2 ≤ ǫ Ak2 for k = 0, . . . , d

• .

Proposition ( A., Gaubert, Marchesini, Tisseur, 2015) Recall that Bǫ = diag(ǫU)Aǫ(ǫ−γX) diag(ǫV ) . Let λ be a simple non-zero eigenvalue of a(γ) = limǫ→0 Bǫ, and Mǫ = Lǫǫγ be a simple eigenvalue of Bǫ such that limǫ→0 Mǫ = λ. Then, we have lim

ǫ→0 κ(Bǫ, Mǫ) = κ(a(γ), λ) .

Theorem ( A., Gaubert, Marchesini, Tisseur, 2015) Denote d := min

i∈[n],wi=0 Ui +

min

j∈[n],zj=0 Vj − max (i,j)∈G(Ui + Vj) .

(1) Then d ≤ 0 and lim inf

ǫ→0 (κ(Aǫ, Lǫ)ǫ−d) > 0 .

In particular, v(κ(Aǫ, Lǫ)) ≥ −d ≥ 0. If z and w have no zero entries and U or V is not a constant vector, then d < 0 and lim inf

ǫ→0 κ(Aǫ, Lǫ) = +∞ .

An example where the diagonal scaling fails:

A =    (X − δ)ǫ2 ǫ ǫ ǫ X − δ ǫ X − δ    =    −δǫ2 ǫ ǫ ǫ −δ ǫ −δ    + X    ǫ2 1 1   

• A has the simple eigenvalues δ and δ ±

√ 2.

• The right and left eigenvectors z and w of A for the eigenvalue δ are

independent of ǫ and satisfy zT = w =

• 1

−1

• .
• limǫ→0 κ(Aǫ, δ) = 2.
• The tropical matrix polynomial associated to A is given by:

A =    −2 −1 −1 −1 −∞ −1 −∞    ⊕ X    −2 −∞ −∞ −∞ −∞ −∞ −∞    .

• It has a unique eigenvalue 1 = 0 with multiplicity 3.
• A(1) has the Hungarian pair (U, V ), with U = V =
• −1
• , and the

graph G = Sat(A(1), U, V ) is fully indecomposable.

• The scaled eigenvalue is Mǫ = δ = Lǫ and the scaled matrix polynomial is

B =    (X − δ) 1 1 1 X − δ 1 X − δ    =    −δ 1 1 1 −δ 1 −δ    + X    1 1 1    .

• The vectors z and w are still right and left eigenvectors of B for the

eigenvalue δ.

• However,

κ(Bǫ, Mǫ) ≥ √ 2 δ . Then, for δ small, κ(Bǫ, Mǫ) is much larger than κ(Aǫ, Lǫ).

Lower bounds on condition numbers

Consider the regular matrix polynomial: A = A0 + XA1 + · · · + X dAd Aj ∈ Cn×n 0 ≤ j ≤ d , Let A = v(A) := A0 ⊕ · · · ⊕ X dAd with Ak = v(Ak) = (|(Ak)ij|)ij, k = 0, . . . , d. Then A is regular. For any finite eigenvalue γ of A, and any Hungarian pair (U, V )

• f log A(γ), we consider the scaled matrix polynomial

B = diag(exp(−U))A(γX) diag(exp(−V )) .

Lower bounds on condition numbers

Theorem ( A., Gaubert, Marchesini, Tisseur, 2015) Let µ be a simple non-zero eigenvalue of B, λ = γµ, and x and w be respectively right and left eigenvectors of B for the eigenvalue µ. Assume x and w have no zero entries. Then κ(A, λ) κ(B, µ) ≥ max(H(exp(U)), H(exp(V ))) 1 n2d T (µ)d H(x) H(w) . where: T (x) := max(|x|, 1/|x|) for x ∈ C, and H(x) := max(|x1|, . . . , |xn|) min(|x1|, . . . , |xn|) for x ∈ Cn.

Lower bounds on condition numbers

Theorem ( A., Gaubert, Marchesini, Tisseur, 2015) Let µ be a simple non-zero eigenvalue of B, λ = γµ, and x and w be respectively right and left eigenvectors of B for the eigenvalue µ. Assume x and w have no zero entries. Then κ(A, λ) κ(B, µ) ≥ max(H(exp(U)), H(exp(V ))) 1 n2d T (µ)d H(x) H(w) . where: T (x) := max(|x|, 1/|x|) for x ∈ C, and H(x) := max(|x1|, . . . , |xn|) min(|x1|, . . . , |xn|) for x ∈ Cn.

Betcke, 2008 introduced a similar scaling, and characterized the optimal scaling in

terms of the true eigenvalues and eigenvectors.

Example continued

Apply the archimedean valuation to Aǫ, for each ǫ fixed: Aǫ := v(Aǫ): Aǫ =    δǫ2 ǫ ǫ ǫ δ ǫ δ    ⊕ X    ǫ2 1 1    =    ǫ2(X ⊕ δ) ǫ ǫ ǫ X ⊕ δ ǫ X ⊕ δ    . For δ < 1, the tropical roots of Aǫ are δ with multiplicity 1 and 1 with multiplicity 2. For γ = δ, Aǫ(γ) has the Hungarian pair (U, V ) given by U =

• log ǫ − log δ

T and V =

• log ǫ

log δ log δ T . Then the diagonal scaling of Aǫ is such that the eigenvalue δ of Aǫ is transformed into the eigenvalue 1 of Bǫ, that the corresponding right and left eigenvectors z and w such that zT = w =

• 1

−1

• , and that

√ 2 + 1 ≤ κ(Bǫ, 1) ≤ √ 7 + 1 .

This work: give majorizations/Ostrowski bounds for the eigen- values of matrix polynomials

• Non-archimedean case
• Archimedean case
• Eigenvectors
• Condition numbers
• Archimedean valuation norms for eigenvalues.

A generalization of P´

• lya inequality

Consider a n × n matrix polynomial A(λ) = A0 + A1λ + · · · + Adλd Aj ∈ Cn×n 0 ≤ j ≤ d , and apply the valuation norm: vN(A)(x) = max

0≤j≤d(Ajxj) ,

with B :=  1 n

n

• i,j=1

|Bij|2  

1 2

. Theorem ( A., Gaubert, Sharify, LAA 2016, and arXiv:1304.2967) Let |λ1| ≥ · · · ≥ |λnd| be the modulus of the eigenvalues of A and γ1 ≥ · · · ≥ γd be the tropical roots of vN(A). Assume det Ad = 0 and let c =

Adn | det Ad|. We have

|λ1 . . . λnk| (α1 . . . αk)n ≤ c(Uk)n , 1 ≤ k ≤ d , with Uk =

• (k+1)k+1

kk

≤

• e(k + 1).

• A lower bound was obtained under technical conditions involving the

condition numbers of the matrices Ak.

• So under restrictive conditions the tropical roots of vN(A) are close to the

tropical eigenvalues of the matrix polynomial v(A).

• However, since the norm is fixed, one cannot construct a diagonal scaling of

the matrix polynomial using only the tropical roots of vN(A).

• So one can only apply a scaling on the eigenvalue, and this is in general not

sufficient to reduce the eigenvalue condition number.

Related results

• Fan, Lin and Van Dooren, 2000 introduced a scaling of the eigenvalues based on

the norms Aj for a matrix quadratic polynomial in order to decrease the backward error.

• Gaubert, Sharify, 2009 introduced a scaling on the eigenvalues based on the

tropical roots of vN(A) and proved some bounds between the eigenvalues of the matrix polynomial A and the tropical roots of vN(A).

• Melman, 2013; Bini, Noferini, Sharify, 2013 and Noferini, Sharify, Tisseur, 2015 :

show a Pellet’s theorem for matrix polynomials A, and compare the bounds with the tropical roots of vN(A).

• However, Pellet’s theorem cannot always be applied, whereas the previous

bounds are (almost) unconditional.

• Bounds for norm valuations applied to block matrices.