QUADRATIC EIGENVALUE PROBLEMS:SOLVENTS AND INVERSE PROBLEMS
Peter Lancaster, University of Calgaryand
Francoise Tisseur, University of Manchester
Spring, 2009.
Lancaster/Tisseur () QEVP Spring, 2009. 1 / 22
Early Gohberg/Lancaster/Rodman Theory
U. of Calgary, Dept of Math and Stat, Research report 419, 1979. (126pp)
Papers in LAA, Canadian J.Math., IEOT, SIMAX, LAMA. 1977-1982.
Gohberg, I., Lancaster, P., and Rodman, L., Spectral analysis of matrixpolynomials, Annals of Math, 1982.
Gohberg, I., Lancaster, P., and Rodman, L., Matrix Polynomials,Academic Press, 1982. (SIAM 2009 - to appear)
Lancaster/Tisseur () QEVP Spring, 2009. 2 / 22
The quadratic problem
Let M, D, K ∈ Cn×n, all Hermitian, with M > 0 and
L(λ) := Mλ2 + Dλ + K .
We will assume that L is monic; M = I .
We will also consider the important special case of D and K real andsymmetric.
Lancaster/Tisseur () QEVP Spring, 2009. 3 / 22
Factorization of L(λ)
In general, do there exist matrices A (a right divisor) and S (a left divisor)such that
Inλ2 + Dλ + K = (Inλ− S)(Inλ− A)
and D∗ = D, K ∗ = K?
YES! But, with eigenvalues “evenly divided” with respect to the real axis.
Can we take advantage of this factorization to solve an inverse problem asfollows?
Assign n eigenvalues and eigenvectors as those of a right divisor A.
Determine a class of compatible left divisors S .
Choose a suitable S
Lancaster/Tisseur () QEVP Spring, 2009. 4 / 22
Factorization of L(λ)
In general, do there exist matrices A (a right divisor) and S (a left divisor)such that
Inλ2 + Dλ + K = (Inλ− S)(Inλ− A)
and D∗ = D, K ∗ = K?YES! But, with eigenvalues “evenly divided” with respect to the real axis.
Can we take advantage of this factorization to solve an inverse problem asfollows?
Assign n eigenvalues and eigenvectors as those of a right divisor A.
Determine a class of compatible left divisors S .
Choose a suitable S
Lancaster/Tisseur () QEVP Spring, 2009. 4 / 22
Preliminaries
Suppose that there are 2r real evs and 2(n − r) non-real ev (in n − rconjugate pairs).
A right divisor immediately determines n of these ev’s and n associatedright eigenvectors, because
(Iλ− A)x = 0, ⇒ L(λ)x = (Inλ− S)(Inλ− A)x = 0.
THE TRIVIAL SOLUTION: We can solve the Hermitian problem bytaking S = A∗. We eschew this solution! Why?
Because real eigenvalues of (Inλ− A∗)(Inλ− A) are necessarily defectiveand
(Inλ− A∗)(Inλ− A) ≥ 0 for all λ ∈ R.
Lancaster/Tisseur () QEVP Spring, 2009. 5 / 22
Preliminaries
Suppose that there are 2r real evs and 2(n − r) non-real ev (in n − rconjugate pairs).
A right divisor immediately determines n of these ev’s and n associatedright eigenvectors, because
(Iλ− A)x = 0, ⇒ L(λ)x = (Inλ− S)(Inλ− A)x = 0.
THE TRIVIAL SOLUTION: We can solve the Hermitian problem bytaking S = A∗. We eschew this solution! Why?
Because real eigenvalues of (Inλ− A∗)(Inλ− A) are necessarily defectiveand
(Inλ− A∗)(Inλ− A) ≥ 0 for all λ ∈ R.
Lancaster/Tisseur () QEVP Spring, 2009. 5 / 22
The Hermitian case
Let S = SR + iSI where SR = 12(S + S∗) and SI = −i
2 (S − S∗) areHermitian.It is not difficult to see that, for a left solvent S ,
SRA− A∗SR =1
2(A2 − (A∗)2). (1)
Then we can prove:
Theorem
Given a matrix A ∈ Cn×n a matrix S ∈ Cn×n is such that both S + A andSA are Hermitian if and only if
S = SR −1
2(A− A∗) = SR − iAI ,
where SR is an Hermitian solution of (1).
When the proposition holds we have, of course,
D = −(S + A) = −(SR + AR), K = SA.
Lancaster/Tisseur () QEVP Spring, 2009. 6 / 22
The Hermitian case
Let S = SR + iSI where SR = 12(S + S∗) and SI = −i
2 (S − S∗) areHermitian.It is not difficult to see that, for a left solvent S ,
SRA− A∗SR =1
2(A2 − (A∗)2). (1)
Then we can prove:
Theorem
Given a matrix A ∈ Cn×n a matrix S ∈ Cn×n is such that both S + A andSA are Hermitian if and only if
S = SR −1
2(A− A∗) = SR − iAI ,
where SR is an Hermitian solution of (1).
When the proposition holds we have, of course,
D = −(S + A) = −(SR + AR), K = SA.Lancaster/Tisseur () QEVP Spring, 2009. 6 / 22
The real-symmetric case
With A ∈ Rn×n write
A = A1 + A2 where A1 =1
2(A + AT ), A2 =
1
2(A− AT ),
and similarly for S .
Corollary
Given a matrix A ∈ Rn×n a matrix S ∈ Rn×n is such that both S + A andSA are real and symmetric if and only if S2 = −A2 and S1 is a realsymmetric solution of
S1A− ATS1 =1
2(A2 − (AT )2). (2)
Lancaster/Tisseur () QEVP Spring, 2009. 7 / 22
Real systems in real arithmetic
A is now to be a real right divisor and we seek real left divisors S . For thespectrum of A write
σ(A) = {σ1, . . . , σ2r} ∪ {ρ1, . . . , ρs} (3)
where 2r + s = n and
σ1 = µ1 + iω1, σ2 = µ2 − iω2, . . . , σ2r = µ2r − iω2r , are distinctnon-real numbers.
ρ1, . . . , ρs are distinct real numbers.
Consider a real Jordan canonical form, JR , for A, assuming that alleigenvalues are simple:
JR = diag
[[µ1 ω1
−ω1 µ1
], . . .
[µr ωr
−ωr µr
], ρ1, . . . , ρs
], (4)
and there is real (e-vector) matrix B such that A = BJRB−1.
Lancaster/Tisseur () QEVP Spring, 2009. 8 / 22
Real systems in real arithmetic
A is now to be a real right divisor and we seek real left divisors S . For thespectrum of A write
σ(A) = {σ1, . . . , σ2r} ∪ {ρ1, . . . , ρs} (3)
where 2r + s = n and
σ1 = µ1 + iω1, σ2 = µ2 − iω2, . . . , σ2r = µ2r − iω2r , are distinctnon-real numbers.
ρ1, . . . , ρs are distinct real numbers.
Consider a real Jordan canonical form, JR , for A, assuming that alleigenvalues are simple:
JR = diag
[[µ1 ω1
−ω1 µ1
], . . .
[µr ωr
−ωr µr
], ρ1, . . . , ρs
], (4)
and there is real (e-vector) matrix B such that A = BJRB−1.Lancaster/Tisseur () QEVP Spring, 2009. 8 / 22
Real systems in real arithmetic
Lemma
If A has n distinct eigenvalues (as in (3)) and B−1AB = JR , then S1 is areal symmetric solution of ZA− ATZ = 0 if and only if S1 = B−TFB−1
where
F = diag
[[a1 b1
b1 −a1
], . . .
[ar br
br −ar
], c1, . . . , cs
], (5)
and the parameters aj , bj , ck are real.
Notice that FT = F and FJR = JTR F = (FJR)T .
Lancaster/Tisseur () QEVP Spring, 2009. 9 / 22
Real systems in real arithmetic
Theorem
Given a matrix A ∈ Rn×n and a real matrix B for which A = BJRB−1 and(4) holds, all real matrices S for which A + S and AS are real andsymmetric have the form S = S1 + S2 where
S1 = A1 + B−TFB−1, S2 = −A2 (6)
and F is an arbitrary real symmetric matrix with the structure of (5).
Lancaster/Tisseur () QEVP Spring, 2009. 10 / 22
Real systems in real arithmetic
Real symmetric system coefficients generated in this way are:
D = −(S + A) = −2A1 − B−TFB−1, (7)
K = SA = ATA + B−T (FJR)B−1. (8)
Clearly, they are determined by the n real parameters defining F in (5).
In this construction the number of real (or non-real) eigenvalues in A canbe doubled (in L(λ)), and there is no further constraint on their positionsin the complex plane.
Lancaster/Tisseur () QEVP Spring, 2009. 11 / 22
Real systems in real arithmetic
Example: Assign
A = BJRB−1 =
2 −2 1 −42 −3 1 −32 −2 −1 −24 −3 1 −5
with eigenvalues −1± i , −2 − 3.If we take
F = diag
[[−2 00 2
], −4, −4
],
we obtain a matching left divisor
S =
−4 4 1 −41 0 1 −50 0 −9 100 −2 13 −22
with ev -28.9022, 1.5426, -4.8432, -2.7971.
Lancaster/Tisseur () QEVP Spring, 2009. 12 / 22
The real symmetric system coefficients obtained are
D =
6 −3 −1 4
0 −1 510 −11
26
, K =
7 −8 −3 13
8 2 −824 −41
99
.
Lancaster/Tisseur () QEVP Spring, 2009. 13 / 22
The Hermitian case
As we are interested in the case of mixed, real and non-real eigenvalues,suppose that
σ(A) = {σ1, . . . , σr} ∪ {ρ1, . . . , ρs} (9)
where r + s = n and
σ1, . . . , σr are distinct non-real numbers,
σj 6= σ̄k for j , k = 1, . . . , r (no conjugate pairs),
ρ1, . . . , ρs are distinct real numbers.
∆ := diag(σ1 · · ·σr ), R := diag(ρ1 · · · ρs). (10)
Lancaster/Tisseur () QEVP Spring, 2009. 14 / 22
The Hermitian case
Then A has the spectral decomposition
A =r∑
j=1
σjPj +s∑
k=1
ρkQk , (11)
where P1, . . . ,Pr ,Q1, . . . Qs form a mutually biorthogonal system ofrank-one projectors onto the eigenspaces of A corresponding to the σj , ρk ,respectively.(If r > 0 then these assumptions do not admit the choice of real matricesA with non-real eigenvalues.)
Lancaster/Tisseur () QEVP Spring, 2009. 15 / 22
The Hermitian case
Theorem
Given a matrix A ∈ Cn×n for which (11) and (10) hold, all matrices S forwhich A + S and AS are Hermitian have the form S = SR + iSI where, forsome diagonal E ∈ Rs×s ,
SR =1
2(A + A∗) + T−∗
[0 00 E
]T−1, SI =
i
2(A− A∗). (12)
This shows that S is determined by A and the s real parameters definingE .
(If A has no real eigenvalues, then the last term in the equation for SR
does not appear and the trivial solution
S =1
2(A + A∗) + i .
i
2(A− A∗) = A∗
is unique.)Lancaster/Tisseur () QEVP Spring, 2009. 16 / 22
Sign charcteristics
In both cases (real/symm. and Hermitian) we can keep track of the signcharacteristics of real eigenvalues.
Lancaster/Tisseur () QEVP Spring, 2009. 17 / 22
Hermitian systems with A ∈ Rn×n
Right divisor Iλ− A be defined by
A =
1 1 02 3 21 1 2
.
Ev of A are 1, 2, 3.(a)If we choose e1 = e2 = e3 = 1, then
S =
24 1 2 1−1 3 10 2 2
35 +3X
j=1
ejYj =
24 3 5/2 3−1/2 7/2 1
2 2 5
35 .
L(λ) has ev 1, 2, 3 with sign characteristic -1.
The ev of S (1.78.., 2.69.., 7.02..) interlace those of A and have signcharacteristic +1.
Lancaster/Tisseur () QEVP Spring, 2009. 18 / 22
(b) If e1 = e2 = e3 = 10, then
S =
24 21 7 114 8 120 2 32
35with ev 42.83..,12.91.., 5.25.., all of +ve type.
This determines a hyperbolic system since all ev are real and, assuming theev’s are ordered, L(λ) has n consecutive ev’s of one type followed by nconsecutive ev’s of the other type with a gap between the nth and n + 1stev’s.
(c) If we let e1 = e2 = 1, e3 = −1, then
S =
24 1 3/2 1−3/2 3 0
0 1 3
35with evs 1.82.., and a conjugate pair: 2.58..± i(1.05..).
Lancaster/Tisseur () QEVP Spring, 2009. 19 / 22
EPILOGUE: Self-adjoint Jordan triples
n × n Hermitian systems: L(λ) =∑l
j=0 Ajλj , Al > 0.
Spectral properties encapsulated in a self-adjoint Jordan triple:
(X , J, PX ∗).
X is n × ln complex (defined by eigenvectors):J is ln × ln complex Jordan:P is ln × ln real: all entries 0 or ±1.(Accounts for sign charcteristics of real eigenvalues.)
⇒ Hermitian moments Γj = X (J j)PX ∗, j = 0, 1, ...⇒ formulae for Hermitian coefficients Aj .
Lancaster/Tisseur () QEVP Spring, 2009. 20 / 22
EPILOGUE: Self-adjoint Jordan triples
n × n Hermitian systems: L(λ) =∑l
j=0 Ajλj , Al > 0.
Spectral properties encapsulated in a self-adjoint Jordan triple:
(X , J, PX ∗).
X is n × ln complex (defined by eigenvectors):J is ln × ln complex Jordan:P is ln × ln real: all entries 0 or ±1.(Accounts for sign charcteristics of real eigenvalues.)
⇒ Hermitian moments Γj = X (J j)PX ∗, j = 0, 1, ...⇒ formulae for Hermitian coefficients Aj .
Lancaster/Tisseur () QEVP Spring, 2009. 20 / 22
Real symmetric systems
Are Hermitian! But have more structure.Spectral properties encapsulated in a real self-adjoint Jordan triple:
(XR , JR , PRXTR ).
XR is n × ln real (defined by eigenvectors):JR is ln × ln real Jordan:PR is ln × ln real: all entries 0 or ±1.(Essentially unchanged.)
⇒ real symmetric moments Γj = X (J j)PX ∗, j = 0, 1, ...⇒ formulae for real symmetric coefficients Aj .
Lancaster/Tisseur () QEVP Spring, 2009. 21 / 22
Real symmetric systems
Are Hermitian! But have more structure.Spectral properties encapsulated in a real self-adjoint Jordan triple:
(XR , JR , PRXTR ).
XR is n × ln real (defined by eigenvectors):JR is ln × ln real Jordan:PR is ln × ln real: all entries 0 or ±1.(Essentially unchanged.)
⇒ real symmetric moments Γj = X (J j)PX ∗, j = 0, 1, ...⇒ formulae for real symmetric coefficients Aj .
Lancaster/Tisseur () QEVP Spring, 2009. 21 / 22
Time to go!
Lancaster/Tisseur () QEVP Spring, 2009. 22 / 22
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