triangularization of quadratic matrix polynomialsftisseur/talks/talk_la12.pdf · extension to...

Research Matters

February 25, 2009

Nick HighamDirector of Research

School of Mathematics

1 / 6

Triangularization ofQuadratic Matrix Polynomials

Françoise TisseurSchool of Mathematics

The University of Manchester

[email protected]://www.ma.man.ac.uk/~ftisseur/

Joint work withYuji Nakatsukasa (M/cr), Leo Taslaman (M/cr)and Ion Zaballa (Universidad del País Vasco)

SIAM Conference on Applied Linear Algebra, June 2012

http://www.ma.man.ac.uk/~ftisseur/

http://www.ma.man.ac.uk

http://www.man.ac.uk

mailto:[email protected]

http://www.ma.man.ac.uk/~ftisseur/

http://www.maths.manchester.ac.uk/~yuji/

http://www.maths.manchester.ac.uk/~taslaman/

Reduction to Triangular Forms

Let A,B ∈ Cn×n and let TA,TB denote triangular matrices.

Single matrices: there exists U ∈ Cn×n such that

U∗AU = TA, U∗U = I, (Schur decomp.).

Pair of matrices: there exist unitary U,V ∈ Cn×n s.t.

U∗AV = TA, U∗BV = TB, (generalized Schur decomp.).

These reductions have many applications (eigenvaluecomputation, matrix function computation, . . . ).

Can we extend these reductions to matrix triples?

MIMS Françoise Tisseur Triangularization 2 / 21

http://www.mims.manchester.ac.uk/

Extension to Matrix Triples

Let Q(λ) = λ2M + λD + K be regular (det Q(λ) 6≡ 0).

Suppose there exist U,V nonsingular s.t.UQ(λ)V = λ2TM + λTD + TK = T (λ) is triangular.

Roots λ(1)j , λ(2)j of λ2(TM)jj + λ(TD)jj + (TK )jj are e’vals

of Q(λ).

Q(λ(k)1 )v1 = U−1T (λ

(k)1 )e1 = 0, k = 1,2.

So λ(1)1 and λ(2)1 must have the same e’vec — a strongcondition!

“We can forget about simultaneous triangularization . . . "Charlie Van Loan, CP3, Monday

Use U(λ), V (λ) unimodular instead!



Equivalences

DefinitionTwo matrix polynomials Q(λ), T (λ) are equivalent if thereare unimodular U(λ),V (λ) s.t. T (λ) = U(λ)Q(λ)V (λ).

Q(λ) and T (λ) have the same finite elementary divisors.

Need equivalence transformations preserving

I the degree (want Q(λ),T (λ) to be quadratic),

I the elementary divisors at infinity (i.e., the elementarydivisors at 0 of revQ(λ) = λ2Q(1/λ) = λ2K + λD + M).



Elementary Divisors at Infinity

L(λ) = λ

0 0 00 0 00 0 1

+

1 0 00 1 00 0 0

has one linear elementary divisors at 0 and two linearelementary at∞. Now1 λ 0

0 1 00 0 1

︸︷︷︸unimodular

L(λ) = λ

0 1 00 0 00 0 1

+

1 0 00 1 00 0 0

,

has one linear elementary divisor at 0 and one elementarydivisor at∞ with partial multiplicity 2.

Equivalences can modify the partial multiplicities at∞.



Triangularization by Strong EquivalencesTheorem (Zaballa & T, 2012)

Any regular Q(λ) = λ2M + λD + K is strongly equivalentover C[λ] to triangular T (λ) = λ2TM + λTD + TK , i.e., thereare unimodular U(λ),V (λ) s.t. T (λ) = U(λ)Q(λ)V (λ) hassame finite and infinite elementary divisors as Q(λ).

I Byproduct of solution to quadratic realizability problem,D.S. Mackey (MS68).

I Proved by Gohberg, Lancaster & Rodman (1982) formonic polynomials of arbitrary degree.

I Extended to regular/singular polynomials of arbitrarydegree by Taslaman, Nakatsukasa, Zaballa & T. (2012).

How to numerically compute T (λ)?MIMS Françoise Tisseur Triangularization 6 / 21


Structure Preserving Transformation (SPT)

Q(λ) = λ2M + λD + K ,

CQ(λ) = λ

[I 00 M

]+

[0 K−I D

].

CQ(λ) CT (λ)

Q(λ) T (λ)(strong) equivalence

U(λ)Q(λ)V (λ)

SPT

SLCQ(λ)SR

lineari-zation

quadrati-zation



A Simple MATLAB Code

Let Q(λ) = λ2I + λD + K be n × n.

A = [zeros(n) -K; eye(n) -D];[U,T] = schur(A,’complex’);X =(U(:,1:2:2*n-1)+U(:,2:2:2*n))/sqrt(2);S = [X A*X];At = S\A*S;



A Simple MATLAB Code

Let Q(λ) = λ2I + λD + K be n × n.

A = [zeros(n) -K; eye(n) -D];[U,T] = schur(A,’complex’);X =(U(:,1:2:2*n-1)+U(:,2:2:2*n))/sqrt(2);S = [X A*X];At = S\A*S;

I AT =

[0 −TK

I −TD

]is in companion form.

I T (λ) = λ2I + λTD + TK : upper triang., Λ(Q) = Λ(T ).

I S is an SPT. Cols of X ∈ C2n×n are orthonormal.



Schur’s Theorem for Complex Matrices

Matrix version: if A ∈ Cn×n then there exists a unitary Usuch that U∗AU = T is a triangular matrix.

Subspaces version:

Theorem

Let A ∈ Cn×n. There are subspaces V1, . . . , Vn of Cn

satisfying(i) Cn = V1 ⊕ V2 ⊕ · · · ⊕ Vn,

(ii) for k = 1 : n, V1 ⊕ · · · ⊕ Vk , is A-invariant,(iii) for k = 1 : n, Vk = 〈uk〉, where u1, . . .un form an

orthonormal system of vectors of Cn.



Schur-like Theorem for Q(λ)

Let Q(λ) = λ2M + λD + K , det M 6= 0.

Theorem (Zaballa, T., 2012)

Let λI − A ∈ C[λ]2n×2n be a linearization of n × n Q(λ).There are subspaces V1, . . . , Vn of C2n satisfying

(i) C2n = V1 ⊕ V2 ⊕ · · · ⊕ Vn,(ii) for k = 1 : n, V1 ⊕ · · · ⊕ Vk is A-invariant,

(iii) for k = 1 : n, dimVk = 2 and Vk = 〈xk ,Axk〉, wherex1, . . . xn form an orthogonal system of vectors of C2n.

I Vk = 〈xk ,Axk〉 is a Krylov subspace of dimension 2.I The xj are generating vectors.I If X = [x1 . . . xn] then S = [X AX ] is nonsingular.



Triangularizing SPT for Q(λ) = λ2M + λD + K

Let S = [X AX ], where X contains generating vectors forlinearization λI − A of Q(λ).

S−1AS =: B ⇐⇒ A[X AX ] = [X AX ]

[B11 B12

B21 B22

].

I

[B11

B21

]=

[0I

]and hence B has companion form.

I 〈x1,Ax1, · · · xk ,Axk〉 A-invariant⇒ B12 and B22 areupper triangular.

S = [X AX ] is a triangularizing SPT.



Schur-like Theorem: Matrix Form

Let Q(λ) = λ2M + λD + K ∈ C[λ]n×n with det(M) 6= 0.

Theorem (Zaballa, T., 2012)

For any linearization λI − A of Q(λ), there exists U ∈ C2n×n

with orthonormal columns s.t. [U AU] is nonsingular and

[U AU]−1A[U AU] =

[0 −T0

In −T1

],

where Inλ2 + T1λ+ T0 is triangular and equivalent to Q(λ).

I The columns of U are generating vectors for A.

I Extends to arbitrary degree matrix polynomials.



Generating Vectors from Schur VectorsTheorem (Nakatsukasa, Taslaman, Zaballa & T, 2012)

Let A ∈ C2n×2n have Schur decomposition

U∗AU =

T11 · · · T1n. . . ...

Tnn

, Tij ∈ C2×2.

If there is vj =

[νj1

νj2

]s.t. Tjjvj 6= αjvj , j = 1 : n then cols of

X = Udiag(v1, . . . , vn) = [ν11u1 + ν12u2, . . . νn1u2n−1 + νn2u2n]are generating vectors.

Hence[X AX ]−1A[X AX ] =

[0 −TK

I −TD

],

where TD,TK are upper triangular.MIMS Françoise Tisseur Triangularization 13 / 21


Schur form with Nonderogatory Blocks

If Tjj 6= αI2, i.e., Tjj is nonderogatory then there exists0 6= vj ∈ C2 s.t. vj is not an e’vec of Tjj .

Theorem (Nakatsukasa, Taslaman, Zaballa, T., 2012)

Any Schur form T of a linearization A ∈ C2n×2n ofQ(λ) ∈ C[λ]n×n is unitarily similar to a Schur form T̃ withnonderogatory 2× 2 diagonal blocks.

I Proof is constructive and nontrivial.

I Relies on the property that λ ∈ Λ(A) = Λ(Q) hasgeometric multiplicity less or equal to n.



Triangular Matrix Coefficients

Q(λ) is equivalent to triangular T (λ) = λ2TM + λTD + TK .

Have explicit expressions for TM ,TD,TK in terms of either

I the n orthonormal generating vectors for A, or

I the Schur form T of A,(or Jordan form of A, see Taslaman MS 35)

where λI − A is a linearization of Q(λ) ∈ C[λ]n×n withnonsingular leading coeff.



Triangular Coefficients from Schur Form

Let λI − A be a linearization of Q(λ) = λ2M + λD + K with

Schur form T =

T11 · · · T1n. . . ...

Tnn

,Tjj ∈ C2×2 nonderogatory.

Take vi ,wi ∈ C2 s.t. ‖vi‖2 = ‖wi‖2 = 1, Tiivi 6= αvi , w∗i vi = 0,

V = diag(v1, . . . , vn) ∈ C2n×n, W = diag(w1, . . . ,wn) ∈ C2n×n.

Q(λ) is equivalent to T (λ) = λ2TM + λTD + TK , where

TM = W ∗TV ,TD = −W ∗T 2V ,TK = −TM((V ∗TV )TD − (V ∗T 2V ))

are triangular.MIMS Françoise Tisseur Triangularization 16 / 21


Triangularizing SPT S

Let S = [X AX ], where X contains generating vectors for

A =

[0 −KM−1

I −DM−1

]so that S−1AS =

[0 −TK

I −TD

]=: AT .

I S−1 = [Y AT Y ] , Y ∈ C2n×n [Garvey et al., 2011].

I Can construct Y from generating vectors X andM,D,K .

No need to factorize (invert) S.



Linear Systems

Let S = [X AX ] = [Y AT Y ]−1 and X =

[X1

X2

], Y =

[Y1

Y2

].

For every λ /∈ Λ(Q),

Q(λ)−1 = (X̃1 + λX2)T (λ)−1(Y1 + λY2)− X2Y2,

where X̃1 = X1 − DX2 + X2TD.



Linear Systems

Let S = [X AX ] = [Y AT Y ]−1 and X =

[X1

X2

], Y =

[Y1

Y2

].

For every λ /∈ Λ(Q),

Q(λ)−1 = (X̃1 + λX2)T (λ)−1(Y1 + λY2)− X2Y2,

where X̃1 = X1 − DX2 + X2TD.

ApplicationsI Solving for x , Q(ω)x = b for many values of|ω| ∈ [ωl , ωh], ωl � ωh.

I Evaluation of transfer function: G(s) = cT Q(s)−1b.



Triangularization over R

Theorem (Zaballa & T, 2012)Q(λ) ∈ R[λ]n×n is triangularizable over R[λ] if and only if

p ≤ n − nc,

where• 2nc is the number of nonreal e’vals and• p is the largest geometric multiplicity of the real e’vals

and e’vals at infinity.

Q(λ) ∈ R[λ]3×3 with elementary divisors in R: (λ− 1),(λ− 1), (λ2 + 1)2 is not triangularizable over R[λ] sincep = 2 > n − nc = 3− 2 = 1.



Quasi-triangular Quadratics

[Zaballa & T, 2012]

Any quadratic Q(λ) ∈ R[λ]n×n is strongly equivalent to aquadratic of the form

T (λ) =

[ 2r 2n−2r

2r T1(λ) T3(λ)2n−2r 0 T2(λ)

],

where r = max{0,p + nc − n} and

I T1(λ) is quasi-triangular with r 2× 2 diag. blocks withelementary divisors (λ− λ1), (λ− λ1), (λ2 + diλ+ ki).Here, λ1 ∈ R has largest geometric multiplicity p.

I T2(λ) is triangular.

(Q(λ) has 2nc nonreal e’vals.)MIMS Françoise Tisseur Triangularization 20 / 21


Summary and Concluding Remarks

I Any regular quadratic is strongly equivalent to atriangular quadratic.

I There is a Schur-like theorem for quadratic matrixpolynomials.

I Triangularizing SPTs are defined by n orthonormal(generating) vectors.

I Can competitive algorithms be designed that computesets of n orthonormal generating vectors?

I Subspace iteration, Krylov subspace methods areworth exploring.



References I

S. D. Garvey, P. Lancaster, A. A. Popov, U. Prells, andI. Zaballa.Filters connecting isospectral quadratic systems.To appear in Linear Algebra Appl., 2012.

Y. Nakatsukasa, L. Taslaman, and F. Tisseur.Reduction of matrix polynomials to simpler forms.Technical report, Manchester Institute for MathematicalSciences, The University of Manchester, UK, 2012.In preparation.



References II

L. Taslaman and F. Tisseur.Triangularization of matrix polynomials.Technical report, Manchester Institute for MathematicalSciences, The University of Manchester, UK, 2012.In preparation.

F. Tisseur and I. Zaballa.Triangularizing quadratic matrix polynomials.MIMS EPrint 2012.30, Manchester Institute forMathematical Sciences, The University of Manchester,UK, 2012.



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