Associating the Invariant Subspaces of a Non-Normal Matrix with Transient Effects in its Matrix Exponential or Matrix Powers
AAssociating the Invariant Subspaces of a Non-Normal Matrix withTransient Effects in its Matrix Exponential or Matrix Powers
Matthew G. Reuter ∗ Department of Applied Mathematics and Statistics & Institute for Advanced Computational Science, Stony Brook University,Stony Brook, New York 11794, United States
Abstract
It is well known that the matrix exponential of a non-normal matrix can exhibit transient growth even whenall eigenvalues of the matrix have negative real part, and similarly for the powers of the matrix when alleigenvalues have magnitude less than 1. Established conditions for the existence of these transient effectsdepend on properties of the entire matrix, such as the Kreiss constant, and can be laborious to use inpractice. In this work we develop a relationship between the invariant subspaces of the matrix and theexistence of transient effects in the matrix exponential or matrix powers. Analytical results are obtained fortwo-dimensional invariant subspaces and Jordan subspaces, with the former causing transient effects whenthe angle between the subspace’s constituent eigenvectors is sufficiently small. In addition to providing afiner-grained understanding of transient effects in terms of specific invariant subspaces, this analysis alsoenables geometric interpretations for the transient effects.
Keywords:
Matrix exponential, matrix powers, transient behavior, non-normal matrices, numerical rangeAMS Codes: 15A16, 15A60
1. Introduction
The matrix exponential e M t , for M ∈ C N × N and t ∈ R (often t ≥ M n ( n ≥
0) also have wide applicability [2]; for instance in Markov processes or in condensed matter physics[3]. When M is normal ( i.e. , M has a complete set of orthogonal eigenvectors), its eigenvalues completelydetermine the behavior of e M t and of M n [2]. That is, if all eigenvalues of M have negative real part, then (cid:107) e M t (cid:107) decays monotonically with t ; likewise, (cid:107) M n (cid:107) decays with n when all eigenvalues of M have magnitudeless than 1.The situation is more complicated when M is not normal [2]. The eigenvalues of M correctly predictasymptotic behaviors as t → ∞ or n → ∞ , but can qualitatively fail for small and intermediate values of t or n . For example, there can be transient growth in e M t such that (cid:107) e M t (cid:107) > < t (cid:28) ∞ eventhough all eigenvalues of M have negative real part. Similar effects can be observed in (cid:107) M n (cid:107) . Numerousapplications of these transient effects — i.e. , (cid:107) e M t (cid:107) > (cid:107) M n (cid:107) > M or the Kreiss constant of M (see chapter 14 of [2]). Second, they do not ∗ Corresponding author ([email protected])
Preprint submitted to Elsevier May 11, 2020 a r X i v : . [ m a t h . SP ] M a y sually provide sharp conditions for when M will or will not display transient effects. Furthermore, betterpredictions of transient effects typically require the more computationally intensive properties of M , e.g. ,the Kreiss constant.The goal of this work is to determine more practical conditions for the existence of transient effects in thematrix exponential or matrix powers of M . Our approach differs significantly from previous studies. Ratherthan relating “holistic” properties of M (such as the Kreiss constant) to transient effects, we examine the k -dimensional ( k ≥
2) invariant subspaces of M [4] and show that transient effects can be associated withinvariant subspaces that satisfy certain conditions. This ultimately provides a more fine-grained explanationof transient effects and potentially introduces a new line of inquiry for characterizing non-normal matricesin terms of their invariant subspaces (see Thm. 4).The main results are stated in Thms. 1 and 2 for e M t and M n , respectively, and are proven in section4. Most of these conditions come from analytical results for the 2-dimensional invariant subspaces of M .The layout of this paper is as follows. First, section 2 introduces our notation and reviews key conceptsfrom matrix analysis. Section 3 then derives conditions for transient effects when M ∈ C × . These resultsare generalized to M ∈ C N × N in section 4, where they apply to the 2-dimensional invariant subspaces of M . Results for higher-dimensional Jordan subspaces are also presented. Additionally, there are severalgeometric interpretations for these results, which are discussed in section 5. We finally offer concludingremarks and ideas for future studies in section 6. Theorem 1 (Transient Effects in the Matrix Exponential) . Let M be a N × N complex matrix ( N ≥ )with all of its eigenvalues in the open left half of C ; that is, Re( λ ) < if λ is an eigenvalue of M . Let S be an invariant subspace of M . Then e M t will have transient effects when at least one of the followingconditions is satisfied. S is a M -dimensional ( M ≥ ) Jordan subspace associated with defective eigenvalue λ and Re( λ ) > − cos (cid:18) πM + 1 (cid:19) . (1a)2. S is a 2-dimensional subspace spanned by eigenvectors associated with eigenvalues λ and λ , λ (cid:54) = λ , θ ∈ (0 , π/ is the angle between the eigenvectors [in the Hermitian sense of Eq. (3) ], and θ < arctan (cid:32) | λ − λ | (cid:112) Re( λ )Re( λ ) (cid:33) . (1b) Theorem 2 (Transient Effects in the Matrix Powers) . Let M be a nonsingular N × N complex matrix( N ≥ ) with all of its eigenvalues in the open unit circle of C ; that is, < | λ | < if λ is an eigenvalueof M . Let S be an invariant subspace of M . Then M n will have transient effects when at least one of thefollowing conditions is satisfied. S is a M -dimensional ( M ≥ ) Jordan subspace. S is a 2-dimensional subspace spanned by eigenvectors associated with eigenvalues λ and λ , λ (cid:54) = λ , θ ∈ (0 , π/ is the angle between the eigenvectors [in the Hermitian sense of Eq. (3) ], and θ < arctan (cid:32) | ln( λ ) − ln( λ ) | (cid:112) ln | λ | ln | λ | (cid:33) . (2) The branch of ln( z ) can always be chosen to make the imaginary part of the numerator of Eq. (2) bein [ − π, π ) .
2. Preliminaries and Notation
Several standard results from matrix analysis will be needed for the discussion and proofs of Thms. 1and 2. We overview them in this section, which also introduces our notation. Throughout this section, we2ssume M ∈ C N × N . The angle between two vectors will be calculated in the Hermitian sense [5] withcos( θ ) = | v · v |(cid:107) v (cid:107) (cid:107) v (cid:107) (3)such that θ ∈ [0 , π/ Every M is unitarily similar to an upper triangular matrix; that is, M = QTQ − , where Q is unitaryand T is upper triangular. T is called a Schur form of M [6]. It is trivial to show that M n = QT n Q − and,by use of the matrix exponential’s Taylor series, e M t = Q e T t Q − [7]. Then, assuming the 2-norm or theFrobenius norm for the matrix norm, we get that (cid:107) e M t (cid:107) = (cid:107) e T t (cid:107) and (cid:107) M n (cid:107) = (cid:107) T n (cid:107) because these matrixnorms are invariant to unitary transformations [6]. It is therefore sufficient to consider a Schur form of M in our analyses. A subspace S ⊆ C N is an invariant subspace of M if M x ∈ S for every x ∈ S [4]. The invariant subspacesof M are closely tied to the Jordan decomposition of M ; every invariant subspace of M is spanned by a set ofeigenvectors and generalized eigenvectors of M . In this way, each invariant subspace of M can be associatedwith eigenvalues of M . An invariant subspace of M spanned by eigenvectors and generalized eigenvectorsthat all belong to the same Jordan chain is called a Jordan subspace of M . A 2- or higher-dimensionalJordan subspace of M is thus associated with a sole defective eigenvalue of M .We will also need the restriction of M onto an invariant subspace S , denoted M | S : S → S and definedby M | S x = M x for x ∈ S . Thus, M | S can be regarded as a dim( S ) × dim( S ) matrix. If f is a well-behaved function, suchas the exponential, then the Jordan form of M and its relation to S shows that [7] f ( M ) | S = f ( M | S ) . The numerical range of M is the compact, convex subset of C given by [8] W ( M ) = (cid:8) x · M x : x ∈ C N , (cid:107) x (cid:107) = 1 (cid:9) . (4)The numerical abscissa, ω ( M ) = max { Re( z ) : z ∈ W ( M ) } , (5)comes from the boundary of W ( M ) and relates to transient effects in e M t [2]. An example of the numericalrange and numerical abscissa is displayed in Figure 1. Transient effects in the matrix exponential e M t occur when all eigenvalues of M have negative real partand (cid:107) e M t (cid:107) > t >
0. Clearly, (cid:107) e M t (cid:107) = 1 when t = 0 such that a sufficient condition for transientbehavior is dd t (cid:13)(cid:13) e M t (cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12) t =0 + > . From Eq. (14.2) in [2], dd t (cid:13)(cid:13) e M t (cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12) t =0 + = ω ( M ) , ( A ) W ( A ) λ λ Figure 1: An example of the numerical range (shaded region) for A in Eq. (10) with λ = − . . i , λ = − . − . i ,and θ = 2 π/
7. The real axis, imaginary axis, and unit circle are shown as solid lines, for reference. The dashed vertical linedepicts Re( z ) = ω ( A ), the numerical abscissa of A . Despite both eigenvalues being in the open left half of C (open unit circle),the numerical range extends into the right half plane (outside the unit circle), which signifies transient effects in the matrixexponential (matrix powers). Thus, a sufficient condition for the existence of transient effects in e M t is ω ( M ) > . (6)Transient effects in the matrix exponential are consequently related to the boundary of W ( M ).Other bounds for predicting the existence of transient effects in e M t can be found in chapter 14 of [2], andinvolve (for example) the pseudospectra of M or the Kreiss constant of M . These quantities are “holistic”,meaning they examine M in its entirety. Our analyses in sections 3 and 4 will look at the relationship betweenspecific invariant subspaces of M and transient effects, thereby providing a more fine-grained analysis oftransient effects.Transient effects in M n are similar. In this case the eigenvalues of M are less than one in magnitude and (cid:107) M n (cid:107) > n >
0. As before, a discussion of transient effects in the matrix powers can be found inchapter 14 of [2], which includes bounds that depend on holistic properties of M .
3. Analytical Results on 2 × Many of the conditions in Thms. 1 and 2 come from analytical results on transient effects in 2 × × C × in a way that facilitates our analyses. The key parameter is theangle between the matrix’s eigenvectors (assuming the matrix is not defective). Finally, sections 3.3 and 3.4derive sufficient conditions for the matrix exponential and matrix powers of a 2 × The numerical range of a 2 × M is a (possibly-degenerate) ellipse with boundary given by Thm.3 [8, 9]. The center of the ellipse is tr( M ) /
2, the eigenvalues of M are the foci of the ellipse, and the major4nd minor axes are stated in Lemma 1. Many of the ensuing results will come from analyzing the geometryof this ellipse, for which an example is displayed in Figure 1. Theorem 3 (Boundary of the Numerical Range of a 2 × . In the complexplane C (cid:39) R , the boundary curve of the numerical range of M ∈ C × is (cid:20) x − Re (cid:18) tr( M )2 (cid:19) , y − Im (cid:18) tr( M )2 (cid:19)(cid:21) S x − Re (cid:16) tr( M )2 (cid:17) y − Im (cid:16) tr( M )2 (cid:17) = det( S )4 , (7) where (cid:20) xy (cid:21) ∈ R , S = (cid:20) (cid:107) M (cid:107) + 2 Re(det( M )) 2 Im(det( M ))2 Im(det( M )) (cid:107) M (cid:107) − M )) (cid:21) , M = M − (tr( M ) / I , and (cid:107) · (cid:107) F is the Frobenius norm. Lemma 1 (Major and Minor Axes of the Numerical Range of a 2 × . The major and minor axes of the elliptical numerical range of M ∈ C × are (cid:113) (cid:107) M (cid:107) + 2 | det( M ) | (8a) and (cid:113) (cid:107) M (cid:107) − | det( M ) | , (8b) respectively, where M = M − (tr( M ) / I , and (cid:107) · (cid:107) F is the Frobenius norm.3.2. Parameterizing × Non-Normal Matrices
Before developing the results for 2 × C × using quantities thatwill facilitate our analysis. Appealing to the Jordan form, there are three cases.First is when the matrix is diagonalizable and has a degenerate eigenvalue. In this case the matrix is ascalar multiple of the identity matrix; it is normal and not of interest to this discussion. We will not considerthis case further.Second is when the matrix is diagonalizable but with distinct eigenvalues, λ and λ . We will denotethis matrix by A . From section 2.1 we can assume without loss of generality that A is upper triangular: A = (cid:20) λ a λ (cid:21) , where a, λ , λ ∈ C and λ (cid:54) = λ . It is straightforward to see that (cid:2) (cid:3) T is a right eigenvector associatedwith λ and that (cid:2) a λ − λ (cid:3) T is a right eigenvector associated with λ . Then, the angle between theeigenvectors [Eq. (3)] is cos( θ ) = | a | (cid:112) | a | + | λ − λ | , (9)which yields | a | = | λ − λ | cot( θ ) . Then, for 0 ≤ ϕ < π , we get A = (cid:20) λ e iϕ | λ − λ | cot( θ )0 λ (cid:21) . (10)The angle between the eigenvectors can thus be used to parameterize the non-normality of A . This ideaof examining the angle between eigenvectors comes from the condition number of an eigenvalue [10], whichessentially considers the angle between the eigenvalue’s left and right eigenvectors.5hird is when the matrix is defective, with sole eigenvalue λ . This matrix will be called D . The Schurform is the 2 × D = (cid:20) λ λ (cid:21) . (11)Some useful properties of A and D are stated in the following Lemmas. Lemma 2.
The major and minor axes of W ( A ) are | λ − λ | csc( θ ) (12a) and | λ − λ | cot( θ ) , (12b) respectively.Proof. The proof is trivial following from Eq. (10) and Lemma 1.
Lemma 3.
The numerical abscissa of A is ω ( A ) = (cid:113) | λ − λ | (cid:0) ( θ ) (cid:1) + Re [( λ − λ ) ]2 √ λ + λ )2 . (13) Proof. S in Thm. 3 is S = | λ − λ | (cid:0) + cot ( θ ) (cid:1) − Re (cid:104) ( λ − λ ) (cid:105) − Im (cid:104) ( λ − λ ) (cid:105) − Im (cid:104) ( λ − λ ) (cid:105) | λ − λ | (cid:0) + cot ( θ ) (cid:1) + Re (cid:104) ( λ − λ ) (cid:105) . Then, recast Eq. (5) as a constrained optimization problem: Maximize x subject to Eq. (7). This can besolved using standard techniques, e.g. , Lagrange multipliers, which produces the result. Lemma 4.
The numerical abscissa of D is ω ( D ) = Re( λ ) + 1 / .Proof. The proof is very similar to that of Lemma 3. The matrix S from Thm. 3 is the identity matrix suchthat the boundary of W ( D ) is ( x − Re( λ )) + ( y − Im( λ )) = 14 . The right-most point of the boundary curve is trivially (Re( λ ) + 1 / , Im( λ )); thus, ω ( D ) = Re( λ ) + 1 / W ( D ) having a radius of1 / × Matrices)
As discussed in section 2.4, e M t has transient behavior when ω ( M ) >
0. We can then examine thenon-defective matrix A in Eq. (10) and the defective matrix D in Eq. (11) using this condition. Lemma 5.
Let A be given as in Eq. (10) with λ (cid:54) = λ and Re( λ ) , Re( λ ) < . Then, e A t will havetransient effects when θ < arctan (cid:32) | λ − λ | (cid:112) Re( λ )Re( λ ) (cid:33) . Proof.
Use Lemma 3 to solve ω ( A ) > θ . 6 orollary 1. Let M ∈ C × have the form (Schur form) M = (cid:20) λ a λ (cid:21) , with λ (cid:54) = λ and Re( λ ) , Re( λ ) < . Then, e M t will have transient effects when | a | > (cid:112) Re( λ )Re( λ ) .Proof. The earlier discussion [Eq. (9)] relates a to θ , the angle between the eigenvectors of M . Usingtrigonometry, θ = arctan (cid:18) | λ − λ || a | (cid:19) , which provides the result when combined with Lemma 5. Lemma 6.
Let D be given as in Eq. (11) with Re( λ ) < . Then, e D t will have transient effects when Re( λ ) > − / .Proof. Use Lemma 4 to solve ω ( D ) > λ ). × Matrices)
Let us now turn our attention to matrix powers M n . Because the existence of M n is complicated forsingular M , we will restrict our attention to nonsingular M . The results here follow straightforwardly fromLemmas 5 and 6 for the matrix exponential. Lemma 7.
Let A be nonsingular and given as in Eq. (10) with λ (cid:54) = λ and < | λ | , | λ | < . Then, A n will have transient effects when θ < arctan (cid:32) | ln( λ ) − ln( λ ) | (cid:112) ln | λ | ln | λ | (cid:33) . Proof.
Because A is not singular, we can write A n = e n ln( A ) by choosing the logarithmic branch cut toavoid both eigenvalues [7]. The result then trivially follows from Lemma 5 by using ln( λ ) and ln( λ ) aseigenvalues. Note that the logarithmic branch cut can always be chosen to make the imaginary part ofln( λ ) − ln( λ ) be in [ − π, π ). Lemma 8.
Let D be nonsingular and given as in Eq. (11) with < | λ | < . Then, D n will have transienteffects.Proof. The proof is similar to that of Lemmas 4 and 7. Because λ (cid:54) = 0, D n = e n ln( D ) . It can be shown that[7] ln( D ) = (cid:20) ln( λ ) λ − λ ) (cid:21) , which leads to S = I / | λ | in Thm. 3. Noting that Re(ln( λ )) = ln | λ | , the boundary of W (ln( D )) is( x − ln | λ | ) + ( y − arg( λ )) = 1 / (4 | λ | ). Thus, ω (ln( D )) = ln | λ | + 12 | λ | . One can verify that ω (ln( D )) > < | λ | <
1, implying that there will always be transient effects.Lemmas 5, 6, 7, and 8 collectively summarize sufficient conditions for M ∈ C × to have transient effectsin its matrix exponential or matrix powers ( M nonsingular assumed for the matrix powers). Note thatLemmas 7 and 8 consider n to be continuous such that transient effects may not be noticeable when n isrestricted to positive integers. Remark.
Although not proven, numerical experiments suggest that the conditions in Lemmas 5, 6, and 7are also necessary for transient effects in e M t or M n when M ∈ C × . Future work should be performed toconfirm or refute this conjecture. . Higher-Dimensional Matrices (Proofs of Thms. 1 and 2) The conditions for transient effects in 2 × N × N matrices by considering the 2-dimensional invariant subspaces of M ∈ C N × N . Theorem4 states the relationship. Theorem 4.
Let M ∈ C N × N and let S ⊆ C N be an invariant subspace of M . If e M | S t admits transientbehavior, then so will e M t . A similar statement holds for ( M | S ) n and M n .Proof. Transient effects in e M t require (cid:107) e M t (cid:107) > t >
0. Because M | S admits transient behavior,we know that 1 < (cid:107) e M | S t (cid:107) for some t >
0. Then, using the definition of the matrix norm,1 < (cid:13)(cid:13)(cid:13) e M | S t (cid:13)(cid:13)(cid:13) = max (cid:107) x (cid:107) =1 (cid:13)(cid:13)(cid:13) e M | S t x (cid:13)(cid:13)(cid:13) = max x ∈ S, (cid:107) x (cid:107) =1 (cid:13)(cid:13) e M t x (cid:13)(cid:13) ≤ max (cid:107) x (cid:107) =1 (cid:13)(cid:13) e M t x (cid:13)(cid:13) = (cid:13)(cid:13) e M t (cid:13)(cid:13) . Thus, if M | S admits transient behavior in the matrix exponential, so must M . Similar logic is used to showthat if ( M | S ) n has transient effects, then M n will as well.Using Thm. 4, Lemma 5 proves the second condition of Thm. 1 and Lemmas 8 and 7 prove the first andsecond conditions of Thm. 2, respectively. Lemma 6 shows the first condition of Thm. 1 for 2-dimensionalJordan subspaces. [12, 13] allow us to consider Jordan subspaces of arbitrary dimension because the numer-ical range is analytically known for these cases. Theorem 5 (Numerical Range of a Jordan Block [12, 13]) . Let J be a M × M Jordan block ( M ≥ ) witheigenvalue λ . Then W ( J ) is all points z ∈ C satisfying | z − λ | ≤ cos (cid:18) πM + 1 (cid:19) . Consequently, ω ( J ) = Re( λ ) + cos (cid:18) πM + 1 (cid:19) . The first condition in Thm. 1 trivially follows from Eq. (6) and Thm. 5. This completes the proofs ofThms. 1 and 2.
5. Geometric Interpretations
The results in section 3 largely came from analyzing a matrix M ∈ C × in terms of the angle between itseigenvectors (assuming M is not defective). This alone leads to new geometric interpretations for transienteffects in the matrix exponential and matrix powers of M , but two other geometric observations can bemade as well.First regards the numerical detection of a defective eigenvalue. Due to numerical instabilities in comput-ing the eigenvalues of a non-normal matrix [2] — especially a defective eigenvalue — it can be very difficultto distinguish ( e.g. ) two eigenvalues that are close to each other from a defective eigenvalue. A recentdiscussion of this issue appears in [14]. Let S be the 2-dimensional invariant subspace of M correspondingto these two eigenvalues. An examination of W ( M | S ) provides insight into the situation. From Lemma 2,the major and minor axes of W ( M | S ) are 2 | λ − λ | csc( θ ) and 2 | λ − λ | cot( θ ), respectively. As θ → + ,meaning M | S approaches a defective matrix, W ( M | S ) will tend toward a circle. Examining the shape ofthe restricted numerical range for the invariant subspace may help determine if it is a Jordan subspace ornot.Second is a relation to antieigenvalue analyses [15]. When M is a self-adjoint, positive-definite matrix(and thus normal), its antieigenvector is the vector that is rotated the most by M . The antieigenvalue is the8orresponding angle of rotation, which will be in [0 , π/ M is 2 ×
2, self-adjoint, and negative-definitewith eigenvalues λ and λ , its antieigenvalue is [15]arcsin (cid:18) | λ − λ || λ + λ | (cid:19) , where the angle of rotation is measured in the Hermitian sense of Eq. (3). Compare this against the criticalangle where ω ( A ) = 0 in Lemma 5, which isarctan (cid:18) | λ − λ | √ λ λ (cid:19) when λ , λ ∈ R and λ , λ <
0. Some basic trigonometry reveals that these two angles are the same;the critical angle for transient effects in e M t is exactly the antieigenvalue of a normal matrix with thesame real eigenvalues. That is, transient effects appear in e M t when the normal matrix with the sameeigenvalues can rotate vectors more than the angle between the eigenvectors of M . Future work mayconsider generalizing this connection to matrices with complex eigenvalues, for which an antieigenvalueanalysis is not well developed.
6. Concluding Remarks
In this work we have established a relationship (Thm. 4) between the invariant subspaces of a non-normalmatrix M ∈ C N × N and transient behavior in e M t ( t >
0) and M n ( n > M , as stated in Thms. 1 and2, respectively. Most of the conditions came from analytic results for 2-dimensional invariant subspaces of M , where the angle between the two associated eigenvectors is a key parameter. We should note that theconditions in Thms. 1 and 2 are sufficient but not necessary for transient effects; it is possible that transienteffects can only be associated with higher-dimensional invariant subspaces.Consequently, one immediate next step is to analyze higher-dimensional invariant subspaces for conditionson transient effects. Although not as straightforward as the case for 2 × × W ( M ) [17, 18]. For instance, whatis the lowest-dimensional invariant subspace S of M that satisfies ω ( M | S ) >
0, thereby inducing transientbehavior? Ideas on the restricted numerical range [19, 20] may help answer this question. Finally, we endwith a practical comment. All of our results make use of the Jordan decomposition of a non-normal M ,which can be difficult to compute accurately [2]. Although the results are theoretically useful, additionalwork needs to be performed to strengthen their practical utility, possibly by further exploring connectionsto the Schur decomposition as in Cor. 1. Acknowledgments
I thank Jay Bardhan, Thorsten Hansen, Dominik Orlowski, Christopher DeGrendele, and JonathanKazakov for helpful conversations.
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