We now show that structured mapping problem naturally arises when analyzing backward errors of approximate eigenelements and approximate invariant subspaces of structured ma- trices. LetS∈ {J,L}.For the rest of this section, we assume thatK=C.LetA∈S.
Now, letX ∈Cn×p be a full column rank matrix, that is, rank(A) = p.LetD ∈Cp×p. We wish to find a matrix 4A ∈ S such that (A+4A)X =XD so thatX is an invariant subspace ofA+4A. Indeed, we wish to find4A that has the smallest norm. Observe that such a matrix4Aexists if and only if4Asatisfies4AX=R,whereR:=XD−AX.Thus the problem of finding4A boils down to a solution of structured mapping problem. For the rest of this section, we denoteηS(X, R) byηS(D, X).Then
ηS(D, X) := inf{k4Ak:4A∈Sand (A+4A)X =XD}. (2.4) For the spectral and the Frobenius norms, we denoteηS(X, D) byηS2(D, X) andηSF(D, X),re- spectively. Note that Theorem 2.2.5 provides a necessary and sufficient condition for existence of4A∈S such that (A+4A)X =XD.Further, for the spectral and the Frobenius norms, Theorem 2.2.6 provides ηS(D, X) and a matrix 4A ∈S such that (A+4A)X =XD and k4Ak=ηS(D, X).SinceX has full rank, we easily obtain thatη2S(D, X) =kR(XHX)−1/2k2
and
ηSF(D, X) =
q
2kR(XHX)−1/2k2F− k(XTX)−1/2XTM R(XHX)−1/2k2F, ifMS∈ {sym,skew-sym}
p2kR(XHX)−1/2k2F− k(XHX)−1/2XHM R(XHX)−1/2k2F, ifMS∈ {Herm,skew-Herm}.
We now consider the special case whenX =x∈CnandD=λ∈Cand derive structured backward errorηS(λ, x).Before we proceed further, we briefly discuss the spectral symmetry of eigenvalues of a structured matrix. Table 2.3 summaries some important spectral symmetries of eigenvalues of structured matrices. These results follow from the fact that whenA∈S,we have eitherM−1ATM =Aor M−1AHM =A.
Recall that (λ, y, x) is an eigentriple ofAifAx=λxandyHA=λyH.Spectral symmetries of structured matrices are also reflected in their eigentriples. Table 2.3 summarizes eigentriples of the structured matrices.
Scalar product yTM x yHM x yTM x
Field K=C K=C K=R
J λ (λ, λ) (λ, λ)
L (λ,−λ) (λ,−λ) (λ,−λ, λ,−λ)
J (λ, M x, x) (λ, M y, x),(λ, y, M x) (λ, M x, x) L (λ, M y, x),(−λ, y, M x) (λ, M y, x),(−λ, y, M x) (λ, M y, x),(−λ, y, M x)
Table 2.3: Spectral symmetry of structured matrices.
Setr:=λx−Ax.Then there is a matrixE∈Ssuch that (A+E)x=λxif and only ifx
andrsatisfy the condition in Table 2.2. Suppose thatxandrsatisfy the condition. Then it follows that η2S(λ, x) =krk2/kxk2andηSF(λ, x) =p
2krk22− |hr, xiM|2≤√
2ηS2(λ, x).
Now by Theorem 2.2.6, definingE by
E=
(xTr)M−1xxH+M−1xrT(I−xxH) +M−1(I−xxT)rxH, ifM A∈sym M−1(I−xxT)rxH−M−1xrT(I−xxH), ifM A∈skew-sym (xHr)M−1xxH+M−1xrH(I−xxH) +M−1(I−xxH)rxH, ifM A∈Herm (xHr)M−1xxH−M−1xrH(I−xxH) +M−1(I−xxH)rxH, ifM A∈skew-Herm.
we have E∈Ssuch that (A+E)x=λxandkEkF =ηFS(λ, x).Further, defining4Aby
4A=
E−xTrM−1(I−xxT)rrT(I−xxH)
krk22− |xHr|2 , ifM A∈sym
E, ifM A∈skew-sym
E−xHrM−1(I−xxH)rrH(I−xxH)
krk22− |xHr|2 , ifM A∈Herm E−xHrM−1(I−xxH)rrH(I−xxH)
krk22− |xHr|2 , ifM A∈skew-Herm.
we have 4A∈Ssuch that (A+4A)x=λxandk4Ak2=ηS2(λ, x).
2.3.1 Structured pseudospectra
Structured pseudospectra provide a convenient framework for analyzing numerics of struc- tured matrices. Let A∈Sbe a structured matrix. Then the structured²-pseudospectrum of A, which we denote byσS²(A),is defined by
σ²S(A) := [
k4Ak≤²
{σ(A+4A) :4A∈S}
whereσ(A) denotes the spectrum ofA.Structured pseudospectra are characterized by struc- tured backward errors of approximate eigenvalues ofA. ForA∈Sandz∈C,define
ηS(z, A) := inf{k4Ak:z∈σ(A+4A), 4A∈S}.
Then it follows that ηS(λ, A) = minkxk2=1{ηS(λ, x) : x ∈ Cn}. We mention that for some z, the infimum may not be attained. In such a case, we setηS(z, A) =∞.Thus ηS(z, A) is the structured backward error of z whenz is treated as an approximate eigenvalue value of A. For the spectral and the Frobenius norms, we denoteηS(λ, A) by ηS2(λ, A) and ηFS(λ, A) respectively.
We set η(λ, A) := σmin(A−λI). Then it is easily seen that η(λ, A) is the unstructured backward error of λas an approximate eigenvalue of A. Unlike in the case of unstructured backward errorη(z, A), determination of structured backward errorηS(z, A) for many inter- esting structures is an open problem. Consequently, determiningσ²S(A) for those structures is an open problem. Note thatηS(z, A)≥η(z, A).For certain structures and for certainz∈C, it turns that ηS(z, A) =η(z, A).
Recall thatM A ∈ {sym,skew-sym,Herm,skew-Herm} when A ∈J ⊂Cn×n or A ∈ L ⊂ Cn×n andM∗=±M,∗ ∈ {T, H}.
Theorem 2.3.1. Let S:={A∈Cn×n:M A∈sym} whenMT =M, andS:={A∈Cn×n : M A∈ {sym,skew-sym}} whenMT =−M. Let A∈S. Then forz ∈C, there exists 4A∈S such that z∈σ(A+4A)andηS(z, A) =k4Ak=η(z, A).Consequently, we have
σS²(A) =σ²(A).
Proof: Consider the case M A∈symand M =MT.Letλ∈C.It follows that A−λI ∈S, that is, (M(A−λI))T =M(A−λI).SinceM(A−λI) is complex symmetric there exists a unitary matrixUsuch that the symmetric Takagi factorization [45]M(A−λI) =UΣUT holds, where Σ is a diagonal matrix containing singular values ofM(A−λI) ordered in descending order of magnitude. Note that η(λ, A) = σmin(A−λI) = σmin(M(A−λI)) = Σ(n, n).Let u:=U(:, n).Then we haveM(A−λI)u=η(λ, A)u.This gives (A−η(λ, A)M−1uuT)u=λu.
Setting 4A:=−η(λ, A)M−1uuT,we have 4A∈Sand k4Ak=η(λ, A) =ηS(λ, A). Hence the results follow.
Next, consider the caseM A∈skew-symandMT =−M.Then forλ∈C,we haveM(A− λI)∈S,that is, (M(A−λI))T =−M(A−λI).SinceM(A−λI) is complex skewsymmetric, we have the skewsymmetric Takagi factorization [45]
M(A−λI) =Udiag(d1,· · · , dm)UT,
where U is unitary, dj :=
"
0 sj
−sj 0
#
, sj ∈ C is nonzero and |sj| are singular values of M(A−λI). Here the blocks dj appear in descending order of magnitude of|sj|. Note that M(A−λI)U = Udiag(d1,· · ·, dm). Let u:= U(:, n−1 : n). Then M(A−λI)u= udm = udmuTu.This gives (M A−udmuT)u=λM u.Hence taking4A:=−M−1udmuT,we have λ∈σ(A+4A), 4A∈S andk4Ak =|sm|=σmin(M(A−λI)) =σmin(A−λI) =η(λ, A).
Hence ηS(λ, A) =η(λ, A) and the desired result follows.
Finally, consider the caseMT =−M andM A∈sym. Letλ∈C. Note thatσmin(M(A− λI)) =σmin(A−λI) =η(λ, A).Letuandvbe unit left and right singular vector ofM(A−λI) corresponding toη(λ, A).ThenM(A−λI)v=η(λ, A)u.This givesAv−η(λ, A)M−1u=λv.
LetE∈Cn×nbe such thatE =ET, Ev=uandkEk= 1.Such a matrix always exists. Then setting 4A :=−η(λ, A)M−1E,we have (A+4A)v =λv,4A∈S and k4Ak=η(λ, A) = ηS(λ, A).Hence the result follows. ¥
For Lie and Jordan algebras corresponding to sesquilinear form induced by M, we have partial equality between structured and unstructured pseudospectra.
Theorem 2.3.2. LetS:={A∈Cn×n:M A∈Herm}whenMH =M andS:={A∈Cn×n: M A ∈skew-Herm} whenMH =−M. Let A∈S. Then forz∈R, there exists 4A∈S such that z∈σ(A+4A)andηS(z, A) =k4Ak=η(z, A).Consequently, we have
σ²S(A)∩R=σ²(A)∩R.
Next, consider S:={A∈Cn×n :M A∈skew-Herm} when MH =M and S:={A∈Cn×n : M A∈Herm} when MH =−M. Let A∈S. Then forz ∈iR, there exists 4A∈S such that
z∈σ(A+4A)andηS(z, A) =k4Ak=η(z, A). Consequently, we have σ²S(A)∩iR=σ²(A)∩iR.
Proof: Consider the caseM A∈HermwhenM =MH.Then forλ∈R, M(A−λI)∈S.Since M(A−λI) is hermitian, we have the spectral decompositionM(A−λI) =Udiag(µ1,· · · , µn)UH, whereU is unitary andµj’s appear in descending order of their magnitudes. Note that|µn|= σmin(M(A−λI)) =σmin(A−λI) =η(λ, A). Now defining4A:=−µnM−1U(:, n)U(:, n)H, we haveλ∈σ(A+4A), 4A∈Sand k4Ak=η(λ, A) =ηS(λ, A).Hence the result follows.
The proof is similar for the case when M A∈skew-HermandMH =−M.
Finally, consider the case whenM A∈skew-HermandMH =M.Then forλ∈iR,the set of purely imaginary numbers, we haveM(A−λI)∈S,that is,M(A−λI) is skew hermitian.
Hence the result follows from spectral decomposition of M(A−λI). The proof is similar for the case when M A∈Herm andMH=−M.¥
Similar results hold for some other important structured matrices such as Toeplitz and Hankel matrices which are not described in the setting of Jordan or Lie algebras.