# A short note on the ratio between sign-real and sign-complex spectral radius of a real square matrix

## A short note on the ratio between sign-real and sign-complex spectral radius of a real square matrix

Linear Algebra and its Applications 529 (2017) 126–132 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.co...

Linear Algebra and its Applications 529 (2017) 126–132

Contents lists available at ScienceDirect

Linear Algebra and its Applications www.elsevier.com/locate/laa

A short note on the ratio between sign-real and sign-complex spectral radius of a real square matrix Florian Bünger Institute for Reliable Computing, Hamburg University of Technology, Am Schwarzenberg-Campus 3, Hamburg 21071, Germany

a r t i c l e

i n f o

Article history: Received 11 April 2017 Accepted 19 April 2017 Available online 24 April 2017 Submitted by V. Mehrmann MSC: 15A42 15A18

a b s t r a c t For a real (n × n)-matrix A the sign-real and the sign-complex spectral radius – invented by Rump – are respectively deﬁned as ρR (A) := max{|λ| | |Ax| = |λx|, λ ∈ R, x ∈ Rn \{0}}, ρC (A) := max{|λ| | |Ax| = |λx|, λ ∈ C, x ∈ Cn \{0}}.

Keywords: Sign-real spectral radius Sign-complex spectral radius Spectral radius Eigenvalue inequalities

For n ≥ 2 we prove ρR (A) ≥ ζn ρC (A) where the constant  1 − cos(π/n) ζn := is best possible. 1 + cos(π/n) © 2017 Elsevier Inc. All rights reserved.

1. Introduction Let n ∈ N := {1, 2, . . . } and K ∈ {R, C}. For a square matrix A ∈ Kn,n the so-called generalized spectral radius is deﬁned by ρK (A) := max{|λ| | |Ax| = |λx|, λ ∈ K, x ∈ Kn \{0}}.1

1

E-mail address: ﬂ[email protected] The equality |Ax| = |λx| is meant componentwise.

http://dx.doi.org/10.1016/j.laa.2017.04.022 0024-3795/© 2017 Elsevier Inc. All rights reserved.

(1)

F. Bünger / Linear Algebra and its Applications 529 (2017) 126–132

127

For K = R it is called sign-real and for K = C sign-complex spectral radius. The generalized spectral radius was invented and investigated by Rump [3–8].2 The new concept was used and extended, for example, by Goldberger and Neumann , Peña , Varga , Zalar , and Zangiabadi and Afshin [11,12]. Rump gave many equivalent deﬁnitions of the generalized spectral radius, e.g. the Max–Min-characterization , Theorem 2.4: ρK (A) =

max

min

x∈Kn \{0} xi =0

|(Ax)i | . |xi |

(2)

Here we are interested in bounding the ratio ρR (A)/ρC (A) between sign-real and sign-complex spectral radius from below independent of a speciﬁc real A, i.e., we are searching for the maximum constant ζn ∈ R≥0 that fulﬁlls ρR (A) ≥ ζn ρC (A) for all A ∈ Rn,n .

(3)

First, note that the one-dimensional case n = 1 is trivial since sign-real and sign-complex spectral radius of a real (1 × 1)-matrix coincide. Therefore we will assume n ≥ 2 from now on. Next, note that bounding the ratio ρR (A)/ρC (A) from above yields 1 as best possible upper bound since by deﬁnition ρR (A) ≤ ρC (A) and ρR (A) = ρC (A) for real diagonal matrices A. In , Theorem 6.1, Rump proved that the skew-symmetric Toeplitz matrix ⎛

0

⎜ ⎜ −1 A := (sign(j − i))1≤i,j≤n = ⎜ . ⎝ . . −1

1 .. . ..

. ...

... .. . .. −1

.

⎞ 1 .. ⎟ .⎟ ⎟ ⎠ 1 0

(4)

has sign-real spectral radius ρR (A) = 1 and sign-complex spectral radius  sin(π/n) = ρ (A) = ρ(A) = A2 = 1 − cos(π/n) C

1 + cos(π/n) =: ζn−1 , 1 − cos(π/n)

(5)

where ρ(A) denotes the common spectral radius, i.e., the maximum modulus of all eigenvalues. Thus ρR (A)/ρC (A) = 1/ρC (A) = ζn for the matrix in (4) and n ≥ 2. The main goal of this short note is to prove that ζn as deﬁned in (5) fulﬁlls (3). This is demonstrated in the following main theorem: Theorem 1. For n ∈ N≥2 and every A ∈ Rn,n we have ρR (A) ≥ ζn ρC (A) 2

R The notation of the sign-real spectral radius changes from ρS 0 (A) in earlier papers to ρ (A).

(6)

F. Bünger / Linear Algebra and its Applications 529 (2017) 126–132

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where  ζn :=

1 − cos(π/n) ∈ 1 + cos(π/n)

1.5 2 , . n n

(7)

The constant ζn is best possible for all n ≥ 2. 2. Proof of Theorem 1 Theorem 1 will follow from the next three lemmas. Lemma 2. For n ∈ N≥2 and ξ := cos

π n

we have ζ :=

1−ξ 1+ξ

 1.5 n

 , n2 .

Proof. For n = 2 we have ξ= 0 and ζ = 1 ∈ [0.75, 1] = [1.5, 2]/n. For n = 3 we have ξ = cos(π/3) = 0.5 and ζ = 1/3 ∈ [1/2, 2/3] = [1.5, 2]/n. Now, let n ≥ 4. The function −2  f : [0, 1] → R, z → 1−z 1+z has derivative f (z) = (1+z)2 < 0 and is therefore monotonically decreasing. For x := π/n the Taylor series expansion of the cosine supplies the ξ-inclusion 2 2 4 ξ1 := 1 − x2 ≤ ξ ≤ 1 − x2 + x24 =: ξ2 . Thus, ζ 2 = f (ξ) ≤ f (ξ1 ) = =

2−

x2 2

=

4 x2

1 = −1

4 π2

1 −

1 n2

·

1 ≤ n2

1 16 4 · ≤ 2 , (8/π)2 − 1 n2 n x2 2

4 π2

1 −

1 16

·

1 n2 (8)

x4 24

1 1 1 · 2 = 4 4 1 n 2− + − 2 2 − 1 π2 n x2 1 − x12 π 2 1 − 12n 2     π2 1 π2 1 π2 π2 2.25 1− · 1 − · 2 ≥ 2 . ≥ ≥ (9) 4 12n2 n2 4 12 · 16 n n

ζ = f (ξ) ≥ f (ξ2 ) = 2

x2 2

x2 2

x4 24

=

Taking square roots, (8) and (9) imply the assertion. 2 Lemma 3. Let ri , si ∈ R2 , i ∈ I := {1, . . . , n}, be a collection of n pairs of vectors in the Euclidean plane. Deﬁne J := {i ∈ I | ri = 0 = si } and suppose that the undirected  angles ωj ∈ [0, π] for j ∈ J between rj and sj fulﬁll j∈J ωj ≤ π. Then there exists a unit vector y ∈ R2 such that (riT y)(sTi y) ≥ 0

for all i ∈ I.

(10)

In other words, each pair ri , si lies either in the left or in the right closed half space with separating line L := {x ∈ R2 | xT y = 0} having normal vector y.

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Proof. Condition (10) clearly holds true for all i ∈ I\J and arbitrary y. The union of the open double cones Cj := {x ∈ R2 | (rjT x)(sTj x) < 0}, j ∈ J, contains non-closed arc   lines on the unit circle S of total arc length j∈J 2ωj ≤ 2π. Thus S  := S\ j∈J Cj = ∅, and any y ∈ S  fulﬁlls (10). 2 Lemma 4. Let A ∈ Rn,n , n ≥ 2, λ ∈ C\{0}, and u, v ∈ Rn such that w = u + iv = 0 and √ |Aw| = |λw|, where i := −1. Then there are α, β ∈ R such that x = αu + βv = 0 and  |Ax| ≥

1 − cos(π/n) · |λx| , 1 + cos(π/n)

(11)

whereby this inequality is meant componentwise. Proof. If u, v are linearly dependent, then w = ηz for some z ∈ {u, v}\{0} and η ∈ C\{0}, so that |Az| = |λz|. Thus (11) holds true for x := z since 1−cos(π/n) 1+cos(π/n) ≤ 1 by Lemma 2. Thus we can assume that u, v are linearly independent. Since λ = 0 we can divide A by |λ| in order to assume also that |λ| = 1. Then Aw = Sw for a complex diagonal matrix S with |S| = I, where I denotes the (n × n)-identity matrix. Decomposing S into real and imaginary part gives S = P + iQ, P = diag(ϕ1 , . . . , ϕn ), Q = diag(ψ1 , . . . , ψn ) ∈ Rn,n , P 2 + Q2 = |S|2 = I, i.e., ϕ2k + ψk2 = 1 for all k = 1, . . . , n. From Au + iAv = Aw = Sw = (P u − Qv) + i(Qu + P v) it follows that M := (Au, Av) = (P u − Qv, Qu + P v) ∈ Rn,2 . Thus, for arbitrary y := (α, β)T ∈ R2 \{0} we have x := (u, v)y = αu + βv ∈ Rn \{0} and Ax = M y. Set ξ := cos nπ and ζ := 1−ξ 1+ξ ∈ (0, 1], see Lemma 2. The inequality |Ax| ≥ ζ|x|, which we have to prove, holds true if, and only if, ((Ax)k + ζxk )((Ax)k − ζxk ) = (Ax)2k − ζ 2 x2k ≥ 0

(12)

for all k = 1, . . . , n. Condition (12) is equivalent to 0 ≤ ((Ax)k + ζxk )((Ax)k − ζxk ) = [(M + ζ(u, v))y]k [(M − ζ(u, v))y]k . For ﬁxed k ∈ {1, . . . , n} let a := ak := (Mk,1 , Mk,2 ) = (ϕk uk − ψk vk , ψk uk + ϕk vk ) be the k-th row of M and b := bk := (uk , vk ) the k-th row of (u, v). Then,

(13)

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a22 = b22 = u2k + vk2 ,

(14)

abT = ϕk (u2k + vk2 ), a ±

ζb22

=

a22

2

b22

(15) ± 2ζab = (1 + ζ ± T

2

2ζϕk )(u2k

+

vk2 ),

(a + ζb)(a − ζb)T = a22 − ζ 2 b22 = (1 − ζ 2 )(u2k + vk2 ).

(16) (17)

Deﬁne r := rk := (a + ζb)T and s := sk := (a − ζb)T and suppose that r = 0 = s. From (16) and (17) it follows that the cosine of the undirected angle ω = ωk ∈ [0, π] between r and s computes as3 cos(ω) =

π  (a + ζb)(a − ζb)T 1 − ζ2 1 − ζ2 . = ≥ = ξ = cos a + ζb2 a − ζb2 1 + ζ2 n (1 + ζ 2 )2 − 4ζ 2 ϕ2k

This implies ω ∈ [0, π/n]. Hence, for J := {k ∈ {1, . . . , n} | rk = 0 =  sk } we proved  2 k∈J ωk ≤ |J| π/n ≤ π. Therefore, Lemma 3 supplies a unit vector y ∈ R such that (13) holds true for all k ∈ {1, . . . , n}. 2 For proving Theorem 1 take some A ∈ Rn,n , n ≥ 2, and choose λ ∈ C and w ∈ Cn \{0} such that |Aw| = |λw| and |λ| = ρC (A). If λ = 0, then clearly (6) holds true. If λ = 0, then Lemma 4 and (2) with K = R also yield (6). The constant ζn is best possible for all n ≥ 2 because the matrix A := (sign(j − i))1≤i,j≤n stated in (4) fulﬁlls (6) with equality.   2 Finally, the inclusion ζn ∈ 1.5 n , n stated in (7) was established in Lemma 2. 2 3. Conclusions Corollary 5. For n ≥ 2 and ζn as deﬁned in (7) we have νn := max{μ ∈ R | ρR (A) ≥ μ ρ(A)

for all A ∈ Rn,n } = ζn ,

(18)

where ρ(A) denotes the common spectral radius of A. Proof. Theorem 1 says ζn = max{μ ∈ R | ρR (A) ≥ μ ρC (A) for all A ∈ Rn,n }. By (1), ρC ≥ ρ holds true so that ζn ≤ νn . On the other hand, the matrix A stated in (4) fulﬁlls ρR (A) = 1 and ρ(A) = ζn−1 , see (5), so that ζn = ρR (A)/ρ(A) ≥ νn . 2 We close with listing sharp enclosures for all ratios x/y, x, y ∈ {ρ, ρR , ρC }, x = y.4 3

This inequality indicates that purely imaginary eigenvalues (P = 0) are somehow the worst case. Since x/y ∈ [a, b] for 0 ≤ a < b ≤ +∞ is equivalent to y/x ∈ [b−1 , a−1 ] with 1/0 := +∞ and 1/ +∞ := 0 it suﬃces to state only one of these reciprocal pairs. 4

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Remark 6. For n ≥ 2 and ζn as deﬁned in (7) the following inclusions hold true: a) b) c)

ρR (A) ∈ [ζn , 1] ρC (A) ρ(A) ∈ [0, ζn−1 ] ρR (A) ρ(A) ∈ [0, 1] ρC (A)

with convention

0 0

for all A ∈ Rn,n , for all A ∈ Rn,n , for all A ∈ Cn,n ,

:= 1. All stated intervals are as tight as possible.

Proof. The stated inclusions follow from ρ, ρR ≤ ρC (see (1)), Theorem 1, Corollary 5, and , Theorem 3.3. The latter says that ρK (A) = 0 for A ∈ Kn,n , K ∈ {R, C}, if, and only if, A is acyclic, i.e., it is permutationally similar to a strictly upper triangular matrix. In particular this means that ρK (A) = 0 implies ρ(A) = 0 and, if A is real, ρR (A) = 0, if, and only if, ρC (A) = 0. Thus, if one of the denominators in a), b), c) is zero, then also the numerator is zero so that by convention the ratio becomes 00 = 1 ≤ ζn−1 , see (7). Furthermore, the matrix A stated in (4) fulﬁlls matrix I satisﬁes ρ(I) = ρR (I) = ρC (I) = 1 so that  A :=

1 1

−1 −1

ρR (A) ρC (A) ρR (I) ρC (I)

ρR (A) ρ(A) and the identity ρ(I) . Finally, for ρC (I)

= ζn = =1=



we have ρC (A) = ρR (A) = 2 and ρ(A) = 0, cf. , p. 15. Padding with zeros this remains ˆ ˆ A) ρ(A) true for the extended (n × n)-matrix Aˆ := diag(A, 0n−2 ) so that ρρ( R (A) ˆ = 0 = ρC (A) ˆ . This shows that all interval bounds in a), b), c) are attained. 2 Acknowledgements I thank Prof. S.M. Rump for fruitful comments and suggestions. References  A. Goldberger, M. Neumann, Perron–Frobenius theory of seminorms: a topological approach, Linear Algebra Appl. 399 (2005) 245–284.  J. Peña, A characterization of the distance to infeasibility under block-structured perturbations, Linear Algebra Appl. 370 (2003) 193–216.  S.M. Rump, Theorems of Perron–Frobenius type for matrices without sign restrictions, Linear Algebra Appl. 266 (1997) 1–42.  S.M. Rump, Almost sharp bounds for the componentwise distance to the nearest singular matrix, Linear Multilinear Algebra 42 (1997) 93–107.  S.M. Rump, Bounds for the componentwise distance to the nearest singular matrix, SIAM J. Matrix Anal. Appl. 18 (1) (1997) 83–103.  S.M. Rump, The sign-real spectral radius and cycle products, Linear Algebra Appl. 279 (1998) 177–180.

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 S.M. Rump, Variational characterizations of the sign-real and the sign-complex spectral radius, Electron. J. Linear Algebra 9 (2002) 112–117.  S.M. Rump, Perron–Frobenius theory for complex matrices, Linear Algebra Appl. 363 (2003) 251–273.  R.S. Varga, Matrix Iterative Analysis, Springer-Verlag, Berlin, Heidelberg, 2009.  B. Zalar, Linear operators preserving the sign-real spectral radius, Linear Algebra Appl. 301 (1999) 99–112.  M. Zangiabadi, H.R. Afshin, A new concept for numerical radius: the sign-real numerical radius, U.P.B. Sci. Bull., Ser. A 76 (3) (2014) 91–98.  M. Zangiabadi, H.R. Afshin, Perron–Frobenius theory on the numerical range for some classes of real matrices, J. Mahani Math. Res. Cent. 2 (2) (2013) 1–15.