On the Similarity to Nonnegative and Metzler Hessenberg Forms Received March 24, 2021; Accepted May 24, 2021

Spec. Matrices 2022; 10:1–8

Research Article Open Access

Christian Grussler and Anders Rantzer* On the similarity to nonnegative and Metzler Hessenberg forms https://doi.org/10.1515/spma-2020-0140 Received March 24, 2021; accepted May 24, 2021

Abstract: We address the issue of establishing standard forms for nonnegative and Metzler matrices by consid- ering their similarity to nonnegative and Metzler Hessenberg matrices. It is shown that for dimensions n ≥ 3, there always exists a subset of nonnegative matrices that are not similar to a nonnegative Hessenberg form, which in case of n = 3 also provides a complete characterization of all such matrices. For Metzler matrices, we further establish that they are similar to Metzler Hessenberg matrices if n ≤ 4. In particular, this provides the rst standard form for controllable third order continuous-time positive systems via a positive controller- Hessenberg form. Finally, we present an example which illustrates why this result is not easily transferred to discrete-time positive systems. While many of our supplementary results are proven in general, it remains an open question if Metzler matrices of dimensions n ≥ 5 remain similar to Metzler Hessenberg matrices.

Keywords: Nonnegative matrices, Hessenberg form, controller-Hessenberg form, positive systems, nonnegative matrix factorization

MSC 2020: 15A21, 15B48, 93C05, 93C28

1 Introduction

Similarity transformation of matrices into structured standard forms is among the most fundamental tools in linear algebra. One of the most common real-valued forms is the upper Hessenberg form, i.e., matrices which are zero except for possible non-zero elements above and on the rst lower diagonal [4]. Non-uniqueness of this form, however, raises the question whether it is possible to incorporate and preserve additional proper- ties. In this work, we consider the following problem:

When is a nonnegative matrix similar to a nonnegative Hessenberg matrix?

We also ask this question when nonnegativity is replaced by Metzler, i.e., matrices that are nonnegative apart R5×5 from their main diagonals. Besides a necessary condition for trace-zero nonnegative matrices in ≥0 [10], little appears to be known about this problem. Rn×n Similar to standard numerical approaches that transform a nonnegative matrix A ∈ ≥0 into upper n−1×n−1 Hessenberg form [4], in this work, we seek a nonnegative similarity transformation T2 ∈ R such that ! ≥0 1 0 the block-diagonal matrix T = yields a nonnegative/Metzler 0 T2 ! ! −1 −1 a11 c2 a11 c2T2 T AT = T T = −1 −1 b2 A2 T2 b T2 A2T2

Christian Grussler: University of California Berkeley, Department of Electrical Engineering and Computer Sciences, Berkeley, CA, E-mail: [email protected] *Corresponding Author: Anders Rantzer: Lund University, Department of Automatic Control, Lund, Sweden, E-mail: [email protected]

Open Access. © 2021 Christian Grussler and Anders Rantzer, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 2 Ë Christian Grussler and Anders Rantzer

−1 Rn−1 −1 with T2 b2 = e1 being the rst canonical unit vector in and T2 A2T2 an upper Hessenberg matrix. As T2 can be found recursively by replacing A with A2, our investigations will start with the case of n = 2 to characterize the case n = 3, which then leads to n = 4. Our main complete characterizations are:

R3×3 I. A ∈ ≥0 is similar to a nonnegative Hessenberg matrix if and only if A does not have a negative eigenvalue of geometric multiplicity 2. II. Every Metzler A ∈ Rn×n with n ≤ 4 is similar to a Metzler Hessenberg matrix.

Rn×n Further, we note that all A ∈ ≥0 with a negative eigenvalue of geometric multiplicity n −1 cannot be similar to a nonnegative Hessenberg matrix. In the language of system and control theory, our approach is equivalent to nding a state-space trans- n−1×n−1 formation T2 ∈ R to a so-called discrete-time (DT) positive system [6] (A2, b2, c2): a state-space system

x(t + 1) = A2x(t) + b2u(t),

y(t) = c2x(t),

−1 −1 with nonnegative system matrices A2, b2, c2; such that (T2 A2T2, T2 b2, c2T2) is a positive system in −1 −1 controller-Hessenberg form [4], i.e., T2 A2T2 is a nonnegative upper Hessenberg matrix, T2 b2 = e1 and c2T2 are nonnegative vectors. In other words, our characterization also attempts to answer the question when a positive systems can be realized in positive controller standard form. So far, this question has only been im- plicitly considered for minimal second-order systems [6]. For continuous-time (CT) positive systems, which are dened by replacing A2 with a Metzler matrix and x(t + 1) with x˙(t), our results rely on a new characterization of all CT positive systems with state-space dimension 3 that can be transformed into a positive system with controller-Hessenberg form. As this also remains true for DT positive systems, where A2 is suciently diagonally dominant, we can further conclude that Item II also applies for suciently diagonally dominant A ∈ R4×4. Unfortunately, as demonstrated by an illustrative example, a general characterization in DT is less trivial. While many of our techniques and tools can be extend to structured cases of larger dimensions, it remains an open question whether a Metzler Hessenberg form also exists for n ≥ 5. The main reason for this analysis limitation is our usage of the fact that polyhedral cones, which are embedded in a 2-dimensional subspace, can be generated by at most two extreme rays. As this is not necessarily true for cones that are embedded in a higher dimensional subspace (see nonnegative rank [2, 7]), our analysis does not simply extend. Finally, note that our results can also be seen towards standard forms for the nonnegative/Metzler inverse eigenvalue problem (see, e.g., [3, 5, 9]), a problem which shares the issue that complete solutions for n > 4 are missing.

2 Preliminaries

n×m n For real valued matrices X = (xij) ∈ R , including vectors x = (xi) ∈ R , we say that X is nonnega- Rn×m R tive, X ≥ 0 or X ∈ ≥0 , if all elements xij ∈ ≥0; corresponding notation are used for positive matrices. Further, X ∈ Rn×n is called Metzler, if all its o-diagonal entries are nonnegative. If X ∈ Rn×n, then σ(X) = {λ1(X), ... , λn(X)} denotes its spectrum, where the eigenvalues are ordered by descending absolute value, i.e., λ1(X) is the eigenvalue with the largest magnitude, counting multiplicity. In case that the magnitude of two eigenvalues coincides, we sub-sort them by decreasing real part. A matrix X is called reducible, if there exists a permutation matrix π ∈ Rn×n such that ! X * πT Xπ = 1 , 0 X2 where X1 and X2 are square matrices. Otherwise, X is irreducible. A spectral characterization of nonnegative matrices is given by the celebrated Perron-Frobenius Theorem [1]. Nonnegative and Metzler Hessenberg forms Ë 3

Rn×n Proposition 1 (Perron-Frobenius). Let A ∈ ≥0 . Then,

i. λ1(A) ≥ 0. ii. If λ1(A) has algebraic multiplicity m0, then A has m0 linearly independent nonnegative eigenvectors related to λ1(A). iii. If A is irreducible, then m0 = 1, λ1(A) > 0 has a positive eigenvectors which is the only nonnegative eigenvector to A. k iv. A is primitive, i.e., there exists a k ∈ N such that A > 0, if and only if A is irreducible with λ1(A) > |λ2(A)|.

n×n X ∈ R is an upper Hessenberg matrix if xij = 0 for all i, j with i > j + 1. The identity matrix of any applicable n size will be denoted by I and its columns, the canonical basic unit vectors in R , are written as e1, ... , en. n×m Using the abbreviation (k1 : k2) = {k1, ... , k2} ⊂ N, we write XI,J for sub-matrices of X ∈ R , where I ⊂ (1 : n) and J ⊂ (1 : j). The element-wise absolute value is written as |X| = (|xij|) and we dene ! X1 0 blkdiag(X1, X2) := 0 X2

n1×n1 n2×n2 n for X1 ∈ R and X2 ∈ R . For a subset S ⊂ R , we write int(S), ∂(S), conv(S) and cone(S) for its interior, boundary, convex hull and convex conic hull. We also use cone(X), X ∈ Rn×m, to denote the convex polyhedral cone that is generated by the columns of X. The set of complex numbers with positive real part is denoted by C>0.

3 Nonnegative and Metzler Hessenberg forms

In this section, we present our main results addressing the questions that were raised in the introduction. Rn×n R Trivially, all A ∈ ≥0 with σ(A) ⊂ ≥0 are similar to a nonnegative Hessenberg matrix, simply by computing its Jordan normal form. Therefore, our investigations will start with σ(A) ⊄ R≥0. We begin by noticing that not every nonnegative matrix is similar to a nonnegative Hessenberg matrix.

Rn×n Proposition 2. For A ∈ ≥0 , the following are equivalent:

i. λ2(A) < 0 has geometric multiplicity n − 1. T Rn ii. A = c(uv − sI), where u, v ∈ >0, 0 ≤ s ≤ mini{ui vi} and c > 0.

Rn×n Proof. i. ⇒ ii.: Let A ∈ ≥0 have λ2(A) < 0 with geometric multiplicity n − 1. Then, A − λ2(A)I = T (λ1(A) − λ2(A))uv , where v and u are left- and right-eigenvectors of A. Since λ1(A) > 0 is the only nonnega- Rn tive eigenvalue of A, A cannot be reducible, which by Proposition 1 implies that u, v ∈ >0. The remaining conditions ensure that A is nonnegative with A ≠ 0. ii. ⇒ i.: Follows by rank(A + csI) = 1 and λ2(A) = −cs. The matrices in Proposition 2 have positive o-diagonal elements, which by their spectral characterization is a property that persists for all nonnegativity preserving similarity transformations. Therefore, for n ≥ 3, these matrices provide a class of nonnegative matrices, which cannot be similar to a nonnegative Hessenberg form. This, however, raises the questions of how fundamental this property is. For the case of n = 3, we will show that these are indeed the only such matrices. To do so, we will characterize when a second-order DT positive system is similar to a DT positive system in controller Hessenberg-form, which will result from the following general lemma that will also be used for the Metzler case later on.

Rn×n Rn Rn×n Lemma 1. Let A ∈ ≥0 be irreducible and b ∈ ≥0 such that Ab ≠ λ1(A)b. Then, there exists a T ∈ ≥0 −1 −1 Rn such that T AT = A and T b ∈ ∂( ≥0). 4 Ë Christian Grussler and Anders Rantzer

Proof. It suces to consider the case with b > 0 and A primitive such that σ(A) ⊂ C>0. If the latter is not fullled, one should replace A with A + kI with suciently large k > 0 in the subsequent discussion. Rn×n −1 First assume that b ∈/ int(cone(A)). Then as lims→∞ cone(A + sI) = ≥0 , the continuity of (A + sI) b in * * * Rn×n −1 s implies the existence of an s ≥ 0 such that b ∈ ∂(cone(A + s I). Hence, T = A + s I ∈ ≥0 with T AT = A −1 Rn and T b ∈ ∂( ≥0). Next, we want show that if Ab ≠ λ1(A)b and b ∈ int(cone(A)), then there exists a smallest k ∈ N such A−k b ∈/ int(cone(A)), meaning that the state-space transformation Ak−1 yields a pair (A, A−k+1b) as in the previous case. To this end, assume the contrary, i.e., A−k b ≥ 0 for all k ∈ N. Then, by Proposition 1, h −k i λ (A)−1 = λ (A−1) < |λ (A−1)| = |λ (A−1)|, which is why 0 = lim min A b − x with V 1 n n−1 2 k→∞ x∈V\{0} kA−k bk being the subspace spanned by the eigenvectors to λ2(A), ... , λn(A) and k·k is an arbitrary norm. This means Rn * Ak x that there exists an x ∈ V ∩ \ {0}. However, then x := limk→∞ k ∈ V is a nonnegative eigenvector ≥0 λ1(A) corresponding to λ1(A), which contradicts that A is irreducible. A direct consequence of Lemma 1 is the following Corollary.

R2 R2×2 Corollary 1. Let (A, b, c) be a second-order DT positive system with b ∈ ≥0\{0}. Then, there exists a T ∈ ≥0 −1 −1 such that (T AT, T b, cT) is a DT positive system in controller Hessenberg-form if and only if Ab ≠ λ1(A)b in case of λ2(A) < 0.

Proof. First note that if b ≯ 0, then A has to be reducible for b to be an eigenvector of A. Hence, it suces to consider b > 0 and we want to nd a p ≥ 0 such that T−1AT ≥ 0 with T = b p ≥ 0. We begin with the case

λ2(A) ≥ 0, which also includes reducible A. Then, Aˆ := A − λ2(A)I ≥ 0, because otherwise there exists an s ≥ 0 such that 0 < (λ1(A) − s)(λ2(A) − s) = det(A − sI) ≤ 0. Hence, since rank(Aˆ) ≤ 1, there exists a p ≥ 0 such that −1 cone(Aˆ) ⊂ cone(T) with det(T) ≠ 0 and, thus, T AT ≥ λ2(A)I ≥ 0. This leaves us with the case, when A is irreducible and λ (A) < 0. If Ab ≠ λ (A)b, then Lemma 1 can be ! 2 1 λ (A) * applied. Otherwise, T−1AT = 1 ≱ 0. 0 λ2(A)

R3×3 We are ready to completely characterize the set of matrices in ≥0 that are similar to nonnegative Hessenberg matrices.

R3×3 Theorem 1. A ∈ ≥0 is not similar to a nonnegative Hessenberg matrix if and only if λ2(A) < 0 has geometric multiplicity 2.

Proof. ⇐: Follows by Proposition 2. R3×3 ⇒: First note that if A ∈ ≥0 has an o-diagonal zero element, then there exists a permutation matrix π ∈ R3×3 such that πTAπ is a nonnegative Hessenberg matrix. Therefore, we can assume that

! T ! A11 vr a11 wl A = T = vl a33 wr A22

˜ R2×2 ˜ −1 ˜ with aij > 0, 1 ≤ i ≠ j ≤ n. Next, if there does not exists a T ∈ ≥0 such that blkdiag(T, 1) A blkdiag(T, 1) or −1 blkdiag(1, T˜) A blkdiag(1, T˜) is nonnegative with at least one o-diagonal zero, then by Corollary 1, vr and T T vl , as well as wr and wl are right- and left eigenvectors to A11 and A22, respectively, and λ2(A11), λ2(A22) < 0. By computing Jordan normal forms of A11 and A22, we can assume that     λ2(A11) 0 0 a11 > 0 0 −1   −1   T1 AT1 =  ≥ 0 λ1(A11) 1  , T2 AT2 =  1 λ1(A22) ≥ 0  (1) 0 > 0 a33 0 0 λ2(A22) with T1 = blkdiag(T11, 1) and T2 = blkdiag(1, T22), and thus λ2(A11), λ2(A22) ∈ σ(A). We want to show now through proof by contradiction that λ2(A) = λ2(A11) = λ2(A22) with geometric multiplicity 2. Assume that Nonnegative and Metzler Hessenberg forms Ë 5

λ2(A11) > λ2(A22). Then, rank(A22 −λ2(A11)I) = 2 and rank(A11 −λ2(A11)I) = 1. Hence, if (A−λ2(A11)I)x = 0, with x ≠ 0, then ! x2 −1 x1 = −x1(A22 − λ2(A11)I) wr = wr . x3 λ2(A11) − λ2(A22) R3 Thus, choosing x1 > 0 implies x ∈ >0, which requires that λ2(A11) ≥ 0 and, therefore, λ2(A11) ≤ λ (A ). Analogously, λ (A ) ≤ λ (A ), which proves that λ (A ) = λ (A ). Finally, let us look into the 2 22 2 22 2 11 2 22 2 11 eigenspace of λ2(A22). We partition T11 = v11 v12 and T22 = v21 v22 , which by (1) implies that ! ! v11 0 and are eigenvectors of A corresponding to the eigenvalue λ2(A11). If these eigenvectors were 0 v22 not linearly independent, we could choose them such that   ! ! 0 v 0 11 = = * ≥ 0, 0 v   22 0 which would require that λ2(A11) ≥ 0 and, therefore, provides a contradiction. Since Metzler matrices allow to freely shift the spectrum, we get the following analogue.

R3×3 R3×3 −1 Corollary 2. For every Metzler A ∈ there exists a T ∈ ≥0 such that T AT is a Metzler Hessenberg matrix.

In order to show that the same holds true in R4×4, we require an extension of Corollary 1 to third-order CT positive systems. We will prove this via the following general lemma.

Rn×n R Rn Lemma 2. Let A ∈ ≥0 be irreducible with σ(A) ⊂ ≥0 and b ∈ >0 be such that Ab = λ1(A)b. Then, there Rn×n −1 −1 exists a T ∈ ≥0 such that T AT ≥ 0 and T b = e1.

Proof. Let V−1AV = J be a Jordan normal form with ones above the diagonal and decreasing diagonal entries, Rn where V = b v2 . . . vn . Then, there exists an α ∈ ≥0 with α1 = 0 such that

T T = b α2b + v2 . . . αn b + vn = V(I + e1α ) ≥ 0.

−1 T T T T Hence, T AT = (I − e1α )J(I + e1α ) = J + e1β with βi = J α + λ1(A)α, i.e., β1 = 0 and βi = αi(λ1(A) − λi(A)) or βi = (λ1(A) − λi(A))αi − αi−1 for i ≥ 2. Since λ1(A) − λi(A) > 0 for all i ≥ 2 by Proposition 1, we can choose a αi−1 suciently large αi ≥ > 0 for the claim to hold. λ1(A)−λi(A))

R3×3 Theorem 2. Let (A, b, c) be a third-order CT positive system. Then, there exists a T ∈ ≥0 such that −1 −1 (T AT, T b, cT) is a CT positive system in controller-Hessenberg form if and only if Ab ≠ λ1(A)b whenever σ(A) ⊄ R. R3×3 C R3×3 Proof. It suces to assume that A ∈ ≥0 and σ(A) ⊂ >0. We want to nd a T = b p q ∈ ≥0 such ! * ˜c that T−1AT = ≥ 0. Then applying Corollary 1 to (A˜ , b˜, ˜c) yields the result. Obviously, such a T cannot b˜ A˜ exist if Ab = λ1(A)b and σ(A) ⊄ R, which is why we want to consider all other cases. Let us rst assume that A is reducible. As in the proof to Corollary 1, Aˆ := A − λ3(A)I ≥ 0 with rank(Aˆ) ≤ 2. Therefore, there exist p, q ≥ 0 such that cone(Aˆ) ⊂ cone({p, q}) ⊂ cone(T) and det(T) ≠ 0, which shows that −1 T AT ≥ λ3(A)I ≥ 0. Let A be irreducible, now. If Ab = λ1(A)b and σ(A) ⊂ R≥0, then the result follows by Lemma 2. Otherwise, by our assumption and Lemma 1, we can assume that b3 = 0, b1, b2 > 0 and that the diagonal entries of A 6 Ë Christian Grussler and Anders Rantzer are suciently large such that   x 0 z 1 1 T   R3×3 Ab := A − bα =  0 y2 z2 =: x y z ≥0 , x3 y3 z3 R3 with α ∈ ≥0 and z1 = 0 or z2 = 0. Hence, Ab is reducible and, thus, as above there exist p, q ≥ 0 such that −1 −1 −1 T T Ab T ≥ 0, which implies that T AT = T Ab T + e1α T ≥ 0.

Theorem 2 is a new characterization of a standard-form for third-order CT positive system. In particular, it implies that every controllable third-order CT positive system has a controller Hessenberg-form, which keeps the system realization positive: this has only be known for minimal second-order systems [6]. As in the proof to Theorem 1, we will use this result to prove our nal characterization.

R4×4 R4×4 −1 Theorem 3. For every Metzler A ∈ there exists a T ∈ ≥0 such that T AT is a Metzler Hessenberg matrix.

Proof. It suces to show that there exists a permutation matrix π ∈ R4×4 such that Theorem 2 can be applied ! ≥0 T Aπ bπ to (Aπ , bπ , cπ), where π Aπ = . Let us assume the contrary. Then, A must be irreducible and for cπ dπ the partition ! T ! A11 vr a11 wl A = T = , vl a44 wr A22 T T it must hold that vr and vl , as well as wr and wl are positive right- and left eigenvectors to irreducible A11 and A22, respectively with σ(A11), σ(A22) ⊄ R. Hence, by computing real diagonal forms to A11 and A22, we can assume that     α −β 0 0 a11 > 0 0 0     −1 β α 0 0  −1  1 λ1(A22) 0 0  T1 AT1 =   , T2 AT2 =   0 0 λ1(A11) 1   0 0 α −β 0 0 > 0 a44 0 0 β α R for some α, β ∈ , T1 = blkdiag(T11, 1), T2 = blkdiag(1, T22), where T11(1:3),{3} = vr and T22(1:3),{1} = −1 −1 ˆ wr. Further, diagonalizing the remaining Metzler blocks (T1 AT1)(3:4),(3:4) and (T1 AT1)(1:2),(1:2) with T11 = R2×2 ˆ R2×2 R2 p * ∈ and T22 = * q ∈ , p, q ∈ >0, respectively, then shows that T T1blkdiag(I, Tˆ11)e3 = vr1p1 vr2p1 vr3p1 p2 and T T2blkdiag(Tˆ22, I)e2 = q1 wr1q2 wr2q2 wr3q2 are positive eigenvectors to λ1(A), which by Proposition 1 and the irreducibility of A are linearly dependent. T Consequently, A11e1 = a11 vr2k vr3k for some k > 0, which is why there exists an s ≥ 0 such ˆ T R3×3 that A11 := A11 − kvr e1 + sI ≥ 0 is reducible. Thus, by Theorem 2, there exists a Tp ∈ ≥0 such that −1 −1 (Tp Aˆ 11Tp , Tp vr , vl Tp) is a CT positive system in controller-Hessenberg form. However, as this also implies −1 that Tp A11Tp is a Metzler Hessenberg matrix, we have a contradiction. Note that while Theorem 2 remains true in DT if A is suciently diagonally dominant, in general this is not true, even if A is similar to a nonnegative Hessenberg matrix. To see this, let us consider     0 0 14 1     A =  0 6 0  = x y z , b = 1 . (2) 15 4 6 0 Nonnegative and Metzler Hessenberg forms Ë 7

0.8 Πˆ (Ab)

0.6 Πˆ (A∞b)

0.4

Πˆ (A2b) 0.2

Πˆ (b) 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 1: Illustration to counter example (2): If there exists a T = (b p q) ≥ 0 with T−1AT ≥ 0, then Πˆ (Ak b) ∈ k conv({Πˆ (b), Πˆ (p), Πˆ (q)}) ⊂ conv({e1, e2}) for all k ∈ N. However, as Πˆ (b), Πˆ (Ab), limk→∞ Πˆ (A b) ∈ ∂(conv({e1, e2})), this is impossible as indicated by the triangle .

The eigenvalues of A are distinct, which by Theorem 1 implies that A is similar to a nonnegative Hessenberg R3×3 −1 R3×3 k matrix. If there exists a T = b p q ∈ ≥0 with T AT ∈ ≥0 , it must hold that A b ∈ cone(T) for all N k N x k ∈ . Equivalently, Π(A b) ∈ conv({Π(b), Π(q), Π(q)}), k ∈ , with Π(x) := P3 denoting the projection i=1 xi R3 P3 onto the simplex simplex S = {x ∈ ≥0 : i=1 xi = 1}. Since S can be mapped bijectively onto conv({e1, e2}), ˆ this is possible if and only if the same applies when replacing Π with Π(x) := Π(x)(2:3). In other words, for T to exist, there has to be a triangle within conv({e1, e2}) that has Πˆ (b) as a corner point and surrounds all ˆ k T R3 Π(A b). That this is impossible can be seen in Fig. 1. In particular, since c ∈ ≥0 can be chosen arbitrarily, even removing the nonnegativity constraint on T will not always allow us to achieve a DT positive system (T−1AT, T−1b, cT) in controller-Hessenberg form. Unfortunately, we do not know whether Theorem 2 extends beyond systems with state-space dimension 3 and as such it is hard to extend Theorem 3 to n > 4. To elaborate on this issue, note that Theorem 2 is based on proving that for all pairs of irreducible A ∈ R3×3 and b ∈ ∂(R3 ) as well as all pairs of reducible A ∈ R3×3 ≥0 ≥0 ≥0 R3 R3×3 R and b ∈ ≥0, there exist T = b t2 t3 , N ∈ ≥0 and s ∈ such that

A − sI = TN. (3)

We were able to establish this result, because rank and nonnegative rank of A−sI coincide if they are bounded by 2 [2]. Extending such a characterization to higher dimensions is a crucial challenges for further advances and lies within the scope of nonnegative matrix factorization [2, 7, 8]. Nonetheless, it is worth pointing out the following variants.

n n×n −1 −1 Proposition 3. Let b ∈ R and A ∈ R be diagonalizable with V AV = D such that |V e1| > 0 and −1 n×n −1 −1 |V b| > 0. Then, there exists a T ∈ R such that T AT = A and T b = e1.

Proof. Let α := V−1e and β := V−1b. Then, T := VEV−1 with E := diag( β1 , ... , βn ) is invertible and T−1b = 1 α1 αn −1 −1 −1 e1. Further, since E DE = D, it follows that T AT = VDV = A. In conjunction with Theorem 3, Proposition 3 yields a partial extension of Theorem 2.

R4 R4×4 R4×4 Corollary 3. Let b ∈ ≥0 and Metzler A ∈ be as in Proposition 3. Then, there exists a T ∈ such that (T−1AT, T−1b) is in CT positive controller-Hessenberg form. 8 Ë Christian Grussler and Anders Rantzer

4 Concluding remarks

We have characterized the similarity of nonnegative and Metzler matrices to nonnegative and Metzler Hessenberg matrices of up to dimension 4, which can be used towards new standard forms for positive systems. As many of the discussed techniques and tools extend beyond these dimensions, such forms can also practically be found for matrices of larger dimensions. In particular, numerical experiments indicate that imposing our derived block-diagonal form for the similarity transformation signicantly helps an alternating projection approach to converge to a feasible invertible transformation. We, further, presented a new related research problem within the scope of nonnegative matrix factorization, where future insights may lead to extensions of our results. Finally, it would be interesting to see how our results can be exploited in the nonnegative/Metzler inverse eigenvalue problem.

Acknowledgement: This work was supported by the ELLIIT Excellence Center at Lund University. It was also supported by the European Research Council under the ERC Advanced Grant ScalableControl n.834142 and by SSF under the grant RIT15-0091 SoPhy.

Data Availability Statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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