Nonconcavity of the Spectral Radius in Levinger's Theorem
NNONCONCAVITY OF THE SPECTRAL RADIUSIN LEVINGER’S THEOREM
LEE ALTENBERGINFORMATION AND COMPUTER SCIENCES,UNIVERSITY OF HAWAI‘I AT M ¯[email protected] E. COHENLABORATORY OF POPULATIONS,ROCKEFELLER UNIVERSITY & DEPARTMENT OF STATISTICS, COLUMBIAUNIVERSITY, NEW YORKDEPARTMENT OF STATISTICS, UNIVERSITY OF [email protected]
Abstract.
Let A ∈ R n × n be a nonnegative irreducible square matrix and let r ( A ) be its spectral radius and Perron-Frobenius eigenvalue. Levinger assertedand several have proven that r ( t ) := r ((1 − t ) A + t A (cid:62) ) increases over t ∈ [0 , / t ∈ [1 / , r ( t ) is concave over t ∈ (0 , A ∈ R × , weighted shift matrices(but not cyclic weighted shift matrices), tridiagonal Toeplitz matrices, and the 3-parameter Toeplitz matrices from Fiedler, but not Toeplitz matrices in general. Ageneral characterization of the range of t , or the class of matrices, for which thespectral radius is concave in Levinger’s homotopy remains an open problem. Dedicated to the memory of Bernard Werner Levinger (1928–2020)
Keywords: circuit matrix, convexity, direct sum, homotopy, nonuniform convergence,skew symmetricMSC2010: 15A18, 15A42, 15B05, 15B48, 15B571.
Introduction
The variation of the spectrum of a linear operator as a function of variation inthe operator has been extensively studied, but even in basic situations like a linearhomotopy (1 − t ) X + t Y between two matrices X , Y , the variational properties of the Date : August 20, 2020.
Linear Algebra and Its Applications doi.org/10.1016/j.laa.2020.07.028 . a r X i v : . [ m a t h . SP ] A ug L. Altenberg & J. E. Cohen spectrum have not been fully characterized. We focus here on Levinger’s theoremabout the spectral radius over the convex combinations of a nonnegative matrix andits transpose, (1 − t ) A + t A (cid:62) .We refer to B ( t ) = (1 − t ) A + t A (cid:62) , t ∈ [0 , Levinger’s homotopy , andthe spectral radius of Levinger’s homotopy as Levinger’s function r ( t ) := r ( B ( t )) = r ((1 − t ) A + t A (cid:62) ).On November 6, 1969, the Notices of the American Mathematical Society receiveda three-line abstract from Bernard W. Levinger for his talk at the upcoming AMSmeeting, entitled “An inequality for nonnegative matrices.”[8] We reproduce it infull:“Theorem. Let A ≥ f ( t ) = p ( tA + (1 − t ) A T ) is a monotone nondecreasing function of t , for 0 ≤ t ≤ /
2, where p ( C ) denotes the spectral radius of the matrix C . This extends a theorem of Os-trowski. The case of constant f ( t ) is discussed.”Levinger presented his talk at the Annual Meeting of the American MathematicalSociety at San Antonio in January 1970. Miroslav Fiedler and Ivo Marek were alsoat the meeting [9]. Fiedler developed an alternative proof of Levinger’s theorem andcommunicated it to Marek [10]. Fiedler did not publish his proof until 1995 [5].Levinger appears never to have published his proof.Marek [10, 11] published the first proofs of Levinger’s theorem, building onFiedler’s ideas to generalize it to operators on Banach spaces. Bapat [1] proveda generalization of Levinger’s theorem for finite matrices. He showed that a nec-essary and sufficient condition for non-constant Levinger’s function is that A havedifferent left and right normalized (unit) eigenvectors ( Perron vectors ) correspondingto the Perron-Frobenius eigenvalue (
Perron root ).Fiedler [5] proved also that Levinger’s function r ( t ) is concave in some open neigh-borhood of t = 1 /
2, and strictly concave when A has different left and right normal-ized Perron vectors. The extent of this open neighborhood was not elucidated.Bapat and Raghavan [2, p. 121] addressed the concavity of Levinger’s function indiscussing “an inequality due to Levinger, which essentially says that for any A ≥ A and A (cid:62) ,is concave.” The inference about concavity would appear to derive from the theoremof [1, Theorem 3] that r ( t A + (1 − t ) B (cid:62) ) ≥ t r ( A ) + (1 − t ) r ( B ) for all t ∈ [0 , A and B have a common left Perron vector and a common right Perron vector. Thesame concavity conclusion with the same argument appears in [13, Corollary 1.17].However, concavity over the interval t ∈ [0 ,
1] would require that for all t, h , h ∈ [0 , r ( t F ( h ) + (1 − t ) F ( h )) ≥ t r ( F ( h )) + (1 − t ) r ( F ( h )), where F ( h ) := h A +(1 − h ) B (cid:62) . While Theorem 3.3.1 of [2] proves this for h = 1 and h = 0, it cannot be Also called Levinger’s transformation [12]. onconcavity of the Spectral Radius in Levinger’s Theorem 3 extended generally to h , h ∈ (0 ,
1) because F ( h ) and F ( h ) (cid:62) will not necessarilyhave common left eigenvectors and common right eigenvectors.Here, we show that the concavity claim is true for 2 × Table 1.
Classes of nonnegative matrices with concave Levinger’sfunction (left), and matrix classes “close” to them with nonconcaveLevinger’s function (right).
Concave Nonconcave × ×
3, 4 × n × n weighted shift matrix Theorem 8 n × n cyclic weighted shift matrix Eq. (6) Matrices that Violate Concavity
A Simple Example.
Let A = / to give B ( t ) = (1 − t ) A + t A (cid:62) = − t t / . (1)The eigenvalues of B ( t ) are { / , + (cid:112) t (1 − t ) , − (cid:112) t (1 − t ) } , plotted in Figure 1. Onthe interval t ∈ [1 / , / r ( B ( t )) = (cid:112) t (1 − t ) is strictly concave. On the intervals t ∈ [0 , /
5] and t ∈ [4 / , r ( B ( t )) is constant. It is clear from the figure that r ( B ( t )) is not concave in the neighborhood of t = 1 / t = 4 / (cid:15) >
0, 12 [ r ( B (1 / − (cid:15) ) + r ( B (1 / (cid:15) )] > r ( B (1 / / . (2)By the continuity of the eigenvalues in the matrix elements [7, 2.4.9], we can make B ( t ) irreducible and yet preserve inequality (2) in a neighborhood of t = 1 / A .The basic principle behind this counterexample is that the maximum of two con-cave functions need not be concave. Here B ( t ) is the direct sum of two block matrices. L. Altenberg & J. E. Cohen !" ! !"$ ! !" λ i t r ( B ( t )) λ = − ! t (1 − t ) λ = ! t (1 − t ) λ = 2 / Figure 1.
Eigenvalues of the matrix B ( t ) from (1), λ = 2 / , λ =+ (cid:112) t (1 − t ) , λ = − (cid:112) t (1 − t ), showing that the spectral radius r ( B ( t ))(thick top line) is not concave around the points t = 0 . t = 0 . t . One block has a constant spectral radius and the otherblock has a strictly concave spectral radius. The spectral radius of B ( t ) is theirmaximum.Another example of this principle is constructed by taking the direct sum of two2 × (cid:18) (cid:19) , but forvalues of t at opposite ends of the unit interval, one 2 × A , with t = 511 / × A , with t = 1 /
8. We take a weighted combination ofthe two blocks with weight h , A ( h ) = (1 − h ) A ⊕ h A , to get: A ( h ) = − h ) − h ) h h . (3)The eigenvalues of B ( t, h ) = (1 − t ) A ( h ) + t A ( h ) (cid:62) are plotted in Figure 2. We seethat there is a narrow region of h below h = 0 . t ∈ [0 , r ( B ( t, h )) = r ((1 − t ) A ( h ) + t A ( h ) (cid:62) ) at h = 0 . t ∈ [0 , A ( h ) may be made irreducible by positiveperturbation of the 0 values without eliminating the nonconcavity.The principle here may be codified as follows. onconcavity of the Spectral Radius in Levinger’s Theorem 5 t h λ i λ λ λ r ( B ( t, . A A λ Figure 2.
Eigenvalues of B ( t, h ) for a two-parameter homotopy:Levinger’s homotopy B ( t, h ) = (1 − t ) A ( h ) + t A ( h ) (cid:62) , t ∈ [0 , A ( h ) = (1 − h ) A ⊕ h A , h ∈ [0 ,
1] (3). The darkband at h = 0 . r ( B ( t, . t where it jumps between the two concave upper mani-folds. Proposition 1.
Let A = A ⊕ A ∈ R n × n , where A and A are irreducible non-negative square matrices. Then r ( t ) := r ((1 − t ) A + t A (cid:62) ) is not concave in t ∈ (0 , if there exists t ∗ ∈ (0 , such that(1) r ((1 − t ∗ ) A + t ∗ A (cid:62) ) = r ((1 − t ∗ ) A + t ∗ A (cid:62) ) ,and(2) dd t r ((1 − t ) A + t A (cid:62) ) (cid:12)(cid:12)(cid:12)(cid:12) t = t ∗ (cid:54) = dd t r ((1 − t ) A + t A (cid:62) ) (cid:12)(cid:12)(cid:12)(cid:12) t = t ∗ .Proof. Let r ∗ := r ( t ∗ ) = r ((1 − t ∗ ) A + t ∗ A (cid:62) ) = r ((1 − t ∗ ) A + t ∗ A (cid:62) ). Since thespectral radius of a nonnegative irreducible matrix is a simple eigenvalue by Perron-Frobenius theory, it is analytic in the matrix elements [14, Fact 1.2]. Thus for eachof A and A , Levinger’s function is analytic in t , and therefore has equal left andright derivatives around t ∗ . So we can set s = d r ((1 − t ) A + t A (cid:62) ) / d t | t = t ∗ and L. Altenberg & J. E. Cohen s = d r ((1 − t ) A + t A (cid:62) ) / d t | t = t ∗ . Then r ((1 − t ∗ − (cid:15) ) A + ( t ∗ + (cid:15) ) A (cid:62) ) = r ∗ + (cid:15)s + O ( (cid:15) ) ,r ((1 − t ∗ − (cid:15) ) A + ( t ∗ + (cid:15) ) A (cid:62) ) = r ∗ + (cid:15)s + O ( (cid:15) ) . For a small neighborhood around t ∗ , r ( t ∗ + (cid:15) ) = r ((1 − t ∗ − (cid:15) ) A + ( t ∗ + (cid:15) ) A (cid:62) )= max (cid:8) r ((1 − t ∗ − (cid:15) ) A + ( t ∗ + (cid:15) ) A (cid:62) ) , r ((1 − t ∗ − (cid:15) ) A + ( t ∗ + (cid:15) ) A (cid:62) ) (cid:9) = r ∗ + (cid:26) (cid:15) min( s , s ) + O ( (cid:15) ) , (cid:15) < ,(cid:15) max( s , s ) + O ( (cid:15) ) , (cid:15) > . A necessary condition for concavity is ( r ( t ∗ + (cid:15) ) + r ( t ∗ − (cid:15) )) ≤ r ( t ∗ ) . However, forsmall enough (cid:15) >
0, letting δ = max( s , s ) − min( s , s ) > r ( t ∗ + (cid:15) ) + r ( t ∗ − (cid:15) )2 = r ∗ + (cid:15) max( s , s ) − min( s , s )2 + O ( (cid:15) )= r ∗ + (cid:15)δ/ O ( (cid:15) ) > r ∗ . The condition for concavity is thus violated. (cid:3)
Toeplitz Matrices.
The following nonnegative irreducible Toeplitz matrix hasa nonconcave Levinger’s function: A = (4)A plot of Levinger’s function for (4) is not unmistakably nonconcave, so instead weplot the second derivative of r ( B ( t )) in Figure 3, which is positive at the boundaries t = 0 and t = 1, and becomes negative in the interior. !" ! ’( ! ’! ! (( t d d t r ( B ( t )) Figure 3.
The second derivative of Levinger’s function for theToeplitz matrix (4). onconcavity of the Spectral Radius in Levinger’s Theorem 7
Weighted Circuit Matrices.
Another class of matrices where Levinger’sfunction can be nonconcave is the weighted circuit matrix. A weighted circuit ma-trix is an n × n matrix in which there are k ∈ [1 , n ] distinct integers i , i , . . . , i k ∈{ , , . . . , n } such that all elements are zero except weights c j , j = 1 , . . . , k , at ma-trix positions ( i , i ) , ( i , i ) , . . . , ( i k − , i k ) , ( i k , i ), which form a circuit. We refer toa positive weighted circuit matrix when the weights are all positive numbers.When focusing on the spectral radius of a positive weighted circuit matrix, we maywithout loss of generality consider its non-zero principal submatrix, whose canonicalpermutation of the indices gives a positive cyclic weighted shift matrix , A , withelements A ij = (cid:26) c i > , j = i mod n + 1 , i ∈ { , . . . , n } , , otherwise.(5)Equation (5) defines a cyclic downshift matrix, while an upshift matrix results fromreplacing j = i mod n + 1 with i = j mod n + 1, which is equivalent for our purposes.Cyclic weighted shift matrices have the form c c
00 0 0 c c . If one of the weights c i is set to 0, the matrix becomes a positive non-cyclic weightedshift matrix. In Section 3.4, we show that Levinger’s function of a positive non-cyclicweighted shift matrix is strictly concave. Cyclicity from a single additional positiveelement c i > re-versible weights , which have been the subject of recent attention [4]. Figure 4 showsLevinger’s function for a 16 ×
16 cyclic weighted shift matrix with two-pivot reversibleweights c j = 16 + sin (cid:18) π j (cid:19) , j = 1 , . . . , . (6)Levinger’s function is convex for most of the interval t ∈ [0 , t = 1 / Matrices with Concave Levinger’s Function
Here we show that several special classes of nonnegative matrices have concaveLevinger’s functions: 2 × L. Altenberg & J. E. Cohen !" t r ( B ( t )) Figure 4.
Nonconcave Levinger’s function for a 16 ×
16 two-pivot re-versible cyclic weighted shift matrix with weights c j = 16+sin(2 πj/ × Matrices.Theorem 2.
Let A ∈ R × be nonnegative and irreducible. Then the spectral radiusand Perron-Frobenius eigenvalue r ( t ) := r ((1 − t ) A + t A (cid:62) ) is concave over t ∈ (0 , ,strictly when A has different left and right Perron-Frobenius eigenvectors.Proof. Let a, b, c, d ∈ (0 , ∞ ) , t ∈ (0 , b (cid:54) = c to assure that A (cid:54) = A (cid:62) andthe left and right Perron-Frobenius eigenvectors are not colinear. Let A := (cid:18) a bc d (cid:19) , B ( t ) := (1 − t ) A + t A (cid:62) . The Perron-Frobenius eigenvalue of B ( t ) is obtained by using the quadratic formulato solve the characteristic equation. After some simplification, r ( t ) := r ( B ( t )) = a + d + (cid:112) ( a − d ) + 4 t (1 − t )( b − c ) + 4 bc . The first derivative with respect to t is r (cid:48) ( t ) = (1 − t ) ( b − c ) (cid:112) ( a − d ) + 4 t (1 − t )( b − c ) + 4 bc . The denominator above is positive for all t ∈ (0 ,
1) because of the assumption that b (cid:54) = c . The second derivative is, again after some simplification, r (cid:48)(cid:48) ( t ) = − b − c ) (( a − d ) + ( b + c ) )(( a − d ) + 4 t (1 − t )( b − c ) + 4 bc ) / < . (7)The numerator in the fraction above is positive because b (cid:54) = c , and the minus sign infront of the fraction guarantees strict concavity for all t ∈ (0 , (cid:3) onconcavity of the Spectral Radius in Levinger’s Theorem 9 Tridiagonal Toeplitz Matrices.Theorem 3 (Tridiagonal Toeplitz Matrices) . Let A ∈ R n × n , n ≥ , be a tridiagonalToeplitz matrix with diagonal elements b ≥ , subdiagonal elements a ≥ , andsuperdiagonal elements c ≥ , with max( a, c ) > . Then for t ∈ (0 , , r ((1 − t ) A + t A (cid:62) ) is concave in t , increasing on t ∈ (0 , / , and decreasing on t ∈ (1 / , , allstrictly when a (cid:54) = c .Proof. The eigenvalues of a tridiagonal Toeplitz matrix A with a, c (cid:54) = 0 are [6, 22-5.18] [3, Theorem 2.4] λ k ( A ) = b + 2 √ ac cos (cid:18) kπn +1 (cid:19) . (8)The matrix (1 − t ) A + t A (cid:62) has subdiagonal values (1 − t ) a + tc and superdiagonalvalues ta + (1 − t ) c . Since at least one of a, c is strictly positive, (1 − t ) a + tc > ta + (1 − t ) c > t ∈ (0 , λ k ( t ) := λ k ((1 − t ) A + t A (cid:62) ), we obtain λ k ( t ) = b + 2 (cid:112) ((1 − t ) a + tc )( ta + (1 − t ) c ) cos (cid:18) kπn +1 (cid:19) . It is readily verified that the first derivatives aredd t λ k ( t ) = cos (cid:18) kπn +1 (cid:19) ( a − c ) (1 − t ) (cid:112) ((1 − t ) a + tc )( ta + (1 − t ) c ) , and the second derivatives ared d t λ k ( t ) = − cos (cid:18) kπn +1 (cid:19) ( a − c ) (cid:2) ((1 − t ) a + tc )( ta + (1 − t ) c ) (cid:3) / . Since (1 − t ) a + tc > ta + (1 − t ) c > t ∈ (0 , a = c both derivatives are identically zero. When a (cid:54) = c , the factorsnot dependent on k are strictly positive for all t ∈ (0 ,
1) except for t = 1 / t ∈ (0 , − t ) a + tc ][ ta + (1 − t ) c ] >
0, there are no inflection points. Therefore each eigen-value is either convex in t or concave in t , depending on the sign of cos( kπ/ ( n + 1)).The maximal eigenvalue is r ( t ) = λ ( t ) = b + 2 (cid:112) ((1 − t ) a + tc )( ta + (1 − t ) c ) cos( π/ ( n +1)) . From its first derivative, since cos( π/ ( n +1)) > r ( t ) is increasing on t ∈ (0 , / t ∈ (1 / , a (cid:54) = c . Since its second derivative isnegative, r ( t ) is concave in t on t ∈ (0 , a (cid:54) = c . (cid:3) Fiedler’s Toeplitz Matrices.
Fiedler [5, p. 180] established this closed for-mula for the spectral radius of a special Toeplitz matrix.
Theorem 4 (Fiedler’s 3-Parameter Toeplitz Matrices) . Consider a Toeplitz matrix A ∈ C n × n , n ≥ , with diagonal values ( v, , . . . , , v, w, u, , . . . , , u ) , with v, w, u ∈ C : A = w u · · · uv w u · · · v w u · · · ... ... . . . . . . . . . ... · · · v w uv · · · v w . (9) Let ω = e πi/n . The eigenvalues of A are λ j +1 ( A ) = w + ω j u (1 − /n ) v /n + ω n − j u /n v (1 − /n ) , j = 0 , , . . . , n − . We apply Theorem 4 to the Levinger function.
Theorem 5.
Let A be defined as in (9) with u, v, w > . Then r ( t ) := r ((1 − t ) A + t A (cid:62) ) is concave in t for t ∈ (0 , , strictly if u (cid:54) = v .Proof. For u, v, w > r ( A ) = λ ( A ) = w + u (1 − /n ) v /n + u /n v (1 − /n ) from Theorem4. Let B ( t ) = (1 − t ) A + t A (cid:62) . Then B ( t ) is again a Toeplitz matrix ofthe form (9), with diagonal values (1 − t ) v + tu, , . . . , , (1 − t ) v + tu, w, (1 − t ) u + tv, , . . . ,
0, (1 − t ) u + tv for matrix elements A i,i + m , with m ∈ { − n, n − } , and i ∈{ max(1 , − m ), . . . , min( n, n − m ) } . So again by Theorem 4, r ( B ( t )) = w + [(1 − t ) u + tv ] (1 − /n ) [(1 − t ) v + tu ] /n + [(1 − t ) u + tv ] /n [(1 − t ) v + tu ] (1 − /n ) . It is readily verified thatd d t r ( B ( t ))= − n − n u v ( u − v ) ( u + v ) × (cid:0) [(1 − t ) v + u ] /n [(1 − t ) u + tv ] (1 − /n ) + [(1 − t ) v + u ] (1 − /n ) [(1 − t ) u + tv ] /n (cid:1) ≤ , with equality if and only if u = v . (cid:3) onconcavity of the Spectral Radius in Levinger’s Theorem 11 With the simple exchange of A n and A n in (9), A would become a circulantmatrix, which has left and right Perron vectors colinear with the vector of all ones, e , and would therefore have a constant Levinger’s function.3.4. Weighted Shift Matrices. An n × n weighted shift matrix, A , has the form A ij = (cid:26) c i , j = i + 1 , i ∈ { , . . . , n − } , , otherwise , where c i are the weights. It is obtained from a cyclic shift matrix be setting any oneof the weights to 0 and appropriately permuting the indices. Unless we explicitly use“cyclic”, we mean non-cyclic shift matrix when we write “shift matrix”.We will show that Levinger’s function for positive weighted shift matrices is strictlyconcave. First we develop some lemmas. Lemma 6.
Let c ∈ C n +1 be a vector of complex numbers and α ∈ C , α (cid:54) = 0 . Then theroots of a polynomial p ( x ) = (cid:80) nk =0 x k α n − k c k are r j = αf j ( c ) , where f j : C n +1 → C , j = 1 , . . . , n .Proof. We factor and apply the Fundamental Theorem of Algebra to obtain p ( x ) = n (cid:88) k =0 x k α n − k c k = α n n (cid:88) k =0 (cid:16) xα (cid:17) k c k = α n n (cid:89) j =1 (cid:16) xα − f j ( c ) (cid:17) . Hence the roots of p ( x ) are { αf j ( c ) | j = 1 , . . . , n } . (cid:3) Lemma 7.
Let α, β ∈ C \ , A ( α, β ) = [ A ij ] be a hollow tridiagonal matrix, where A ij > for j = i + 1 and j = i − , A ij = 0 otherwise, and A ij = (cid:26) α c ij , j = i + 1 , i ∈ { , . . . , n − } ,β c ij , j = i − , i ∈ { , . . . , n } , so A ( α, β ) has the form A ( α, β ) = α c · · · β c α c · · · β c . . . ... ... . . . . . . . . . ... ... . . . α c n − ,n −
00 0 0 · · · β c n − ,n − α c n − ,n · · · β c n,n − . Let c ∈ C n − represent the vector of c ij constants.Then the eigenvalues of A are of the form √ αβ f h ( c ) , h = 1 , . . . , n , where f h : C n − → C are functions of the c ij constants that do not depend on α or β . Proof.
The characteristic polynomial of A is p A ( λ ) = det( λ I − A )= (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ − α c · · · − β c λ − α c · · · − β c λ . . . 0 0 0... ... . . . . . . . . . ... ...0 0 0 . . . λ − α c n − ,n −
00 0 0 · · · − β c n − ,n − λ − α c n − ,n · · · − β c n,n − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The characteristic polynomial has the recurrence relation p A k ( λ ) = λ p A k − ( λ ) − α β c k,k − c k − ,k p A k − ( λ ) , k ∈ { , . . . , n } , (10)with initial conditions p A ( λ ) = λ − α β c c , and(11) p A ( λ ) = λ, (12)where A k is the principal submatrix of A over indices 1 , . . . , k .We show by induction that for all k ∈ { , . . . , n } , p A k ( λ ) = k (cid:88) j =0 λ j ( αβ ) ( k − j ) / g jk ( c ) = k (cid:88) j =0 λ j (cid:112) αβ ( k − j ) g jk ( c ) , (13)where each g jk : C n − → C , k ∈ { , . . . , n } , j ∈ { , . . . , k } , is a function ofconstants c .From (11), we see that (13) holds for k = 2: p ( A )( λ ) = λ − α β c c . For k = 3, from the recurrence relation (10) and initial conditions (12), (11), wehave p ( A )( λ ) = λ p A ( λ ) − αβ c c p A ( λ ) = λ ( λ − αβ c c ) − αβ c c λ = λ − λ (cid:112) αβ ( c c + c c ) , which satisfies (13). These are the basis steps for the induction. onconcavity of the Spectral Radius in Levinger’s Theorem 13 For the inductive step, we need to show that if (13) holds for k − , k − k . Suppose that (13) holds for 2 ≤ k − , k − ≤ n −
1. Then p A k ( λ ) = λ p A k − ( λ ) − αβ c k,k − c k − ,k p A k − ( λ )= λ k − (cid:88) j =0 λ j (cid:112) αβ ( k − − j ) g j,k − ( c ) − αβ c k,k − c k − ,k k − (cid:88) j =0 λ j (cid:112) αβ ( k − − j ) g j,k − ( c )= k (cid:88) j =1 λ j (cid:112) αβ ( k − j ) g j − ,k − ( c ) − k − (cid:88) j =0 λ j (cid:112) αβ ( k − j ) c k,k − c k − ,k g j,k − ( c ) , which satisfies (13). Thus by induction p A n ( λ ) satisfies (13).Then Lemma 6 implies that the parameters { α, β } appear as the linear factor √ αβ in each root of the characteristic polynomial of A ( α, β ) — its eigenvalues. (cid:3) Theorem 8 (Weighted Shift Matrices) . Levinger’s function is strictly concave fornonnegative weighted shift matrices with at least one positive weight.Proof.
Let the positive weighted shift matrix A be defined as A ij = (cid:26) c i ≥ , j = i + 1 , i ∈ { , . . . , n − } , , otherwise , where c i are the weights and c i > i = 1 , . . . , n − B ( t ) = (1 − t ) A + t A (cid:62) are of the form λ i ( B ( t )) = (cid:112) t (1 − t ) f i ( c ), where c is the vector of weights, and f i : R n − → R , since B ( t ) is a direct sum of one or more (if some c i = 0) Jacobimatrices and these have real eigenvalues [6, 22.7.2].If at least one weight c i is positive, then B ( t ) has a principal submatrix (cid:18) − t ) c i t c i (cid:19) with a positive spectral radius for t ∈ (0 , r ( B ( t )) > t ∈ (0 , t ∈ (0 , r ( B ( t )) = λ ( B ( t )) = (cid:112) t (1 − t ) f ( c ) >
0. Since (cid:112) t (1 − t ) is strictly concave in t for t ∈ (0 , t for t ∈ (0 , (cid:3) Corollary 9.
Levinger’s function is strictly concave for a nonnegative hollow tridi-agonal matrix, A ∈ R n × n , in which A ii = 0 for i ∈ { , . . . , n } , and where for each i ∈ { , . . . , n − } , A i,i +1 A i +1 ,i = 0 , and for at least one i , A i,i +1 > .Proof. A is derived from a weighted shift matrix by swapping some elements ofthe superdiagonal A i,i +1 to the transposed position in the subdiagonal, A i +1 ,i . Thedeterminant of Levinger’s homotopy det( λ I − B ( t )) = det( λ I − (1 − t ) A − t A (cid:62) )remains unchanged under such swapping because the term αβ c k,k − c k − ,k in (10), which is (1 − t ) t c k − ,k in the weighted shift matrix, remains invariant under swappingas t (1 − t ) c k,k − . (cid:3) We complete the connection to positive weighted circuit matrices with this corol-lary.
Corollary 10.
By setting one or more, but not all, of the weights in a positiveweighted circuit matrix to , Levinger’s function becomes strictly concave.Proof. A positive weighted circuit matrix where some but not all of the positiveweights are changed to 0 is, under appropriate permutation of the indices, a nonneg-ative weighted shift matrix to which Theorem 8 applies. (cid:3)
What kind of transition does Levinger’s function make during the transition froma cyclic weighted shift matrix with nonconcave Levinger’s function to a weightedshift matrix with its necessarily concave Levinger’s function, as one of the weights islowered to 0? Does the convexity observed in Figure 4 at the boundaries t = 0 and t = 1 flatten and become strictly concave for some positive value of that weight? Weexamine this transition for the cyclic shift matrix in example (6) (Figure 4). Theminimal weight is c = 16 + sin (cid:0) π (cid:1) = 15. Figure 5 plots Levinger’s function as c is divided by factors of 2 .Figure 6 plots the second derivatives of Levinger’s function. We observe non-uniform convergence to the c = 0 curve. As c decreases, the second derivativeconverges to the c = 0 curve over wider and wider intervals of t , but outside ofthese intervals the second derivative diverges from the c = 0 curve, attaining largervalues near and at the boundaries t = 0 and t = 1 with smaller c . Meanwhile for c = 0, Levinger’s function is proportional to (cid:112) t (1 − t ), the second derivative ofwhich goes to −∞ as t goes to 0 or 1. When c > B (0) and B (1) are irreducible,and when c = 0, B ( t ) is irreducible for t ∈ (0 , c = 0, B (0) and B (1)are reducible matrices. While the eigenvalues are always continuous functions of theelements of the matrix, the derivatives of the spectral radius need not be, and in thiscase, we see an unusual example of nonuniform convergence in the second derivativeof the spectral radius.4. Matrices with Constant Levinger’s Function
Bapat [1] and Fiedler [5] identified matrices with colinear left and right Perronvectors as having constant Levinger’s function. Here we make explicit a propertyimplied by this constraint that appears not to have been described. We use thecentered representation of Levinger’s homotopy. The symmetric part of a squarematrix A is S ( A ) := ( A + A (cid:62) ) / . (14) onconcavity of the Spectral Radius in Levinger’s Theorem 15 !" − − − − t r ( B ( t )) c = 15 × : Figure 5.
Levinger’s function for the cyclic weighted shift matrixfrom (6) in the limit as weight c goes toward 0 by being multipliedby successive powers of 2 − . The topmost line with c = 15 × !" ! %!! ! $!! ! c = 15 × : 1 2 − − − − d d t r ( B ( t )) t Figure 6.
The second derivative of Levinger’s function for the cyclicweighted shift matrix from (6) as weight c goes toward 0 by beingmultiplied by successive powers of 2 − .The skew symmetric part of A is K ( A ) := ( A − A (cid:62) ) / . (15)Then A = S ( A ) + K ( A ). Levinger’s homotopy in this centered representation isnow, suppressing the A argument, C ( p ) := S + p K , p ∈ [ − , , and Levinger’s function is c ( p ) := r (( p + 1) /
2) = r ( S + p K ) . The range of p in this centered representation may be extended beyond [ − , C ( p ) ≥ , to the interval p ∈ [ − α, α ] where α = min i,j A ij + A ji | A ji − A ij | ≥ . Theorem 11.
Let A ∈ R n × n be irreducible and nonnegative. Then r ((1 − t ) A + t A (cid:62) ) is constant in t ∈ [0 , if and only if the Perron vector of A + A (cid:62) is in the null spaceof A − A (cid:62) .Proof. [1] and [5] proved that r ((1 − t ) A + t A (cid:62) ) is constant in t ∈ [0 ,
1] if and only ifthe left and right Perron vectors of A are colinear. Suppose the left and right Perronvectors of A are colinear. Without loss of generality, they can be normalized to sumto 1 in which case they are identical. Let the left and right Perron vectors of A be x . Then 12 ( A + A (cid:62) ) x = r ( A ) x , and ( A − A (cid:62) ) x = r ( A ) ( x − x ) = . Hence x is the Perron vector of A + A (cid:62) and x > is in the null space of A − A (cid:62) .For the converse, let the Perron vector of A + A (cid:62) be x > , and let x be in thenull space of A − A (cid:62) . Then( A + A (cid:62) ) x = r ( A + A (cid:62) ) x and ( A − A (cid:62) ) x = Ax − A (cid:62) x = , which gives Ax = 12 [( A + A (cid:62) ) + ( A − A (cid:62) )] x = 12 r ( A + A (cid:62) ) x + = r ( A + A (cid:62) )2 x and A (cid:62) x = 12 [( A + A (cid:62) ) − ( A − A (cid:62) )] x = 12 r ( A + A (cid:62) ) x − = r ( A + A (cid:62) )2 x hence x is a Perron vector of A and of A (cid:62) . (cid:3) Corollary 12.
Let S = S (cid:62) ∈ R n × n be a nonnegative irreducible symmetric matrix,and K = − K (cid:62) ∈ R n × n be a nonsingular skew symmetric matrix such that A = S + K ≥ . Then n is even and A has a non-constant Levinger’s function. onconcavity of the Spectral Radius in Levinger’s Theorem 17 Proof. If K is a nonsingular skew symmetric matrix, n must be even, since odd-orderskew symmetric matrices are always singular [6, 2-9.27]. If C ( p ) := S + p K with K nonsingular, then because the null space of K is { } , C ( p ) must have a non-constantLevinger’s function c ( p ) by Theorem 11. (cid:3) The following corollary pursues the observation made by an anonymous reviewerthat a matrix A with colinear left and right Perron vectors is orthogonally similarto a direct sum (cid:0) r ( A ) (cid:1) ⊕ F for some square matrix F . This entails that the skewsymmetric part of A is orthogonally similar to (cid:0) r ( A ) − r ( A ) (cid:1) ⊕ ( F − F (cid:62) ) / (cid:0) (cid:1) ⊕ ( F − F (cid:62) ) /
2, and is thus singular.
Corollary 13.
Let S = S (cid:62) ∈ R n × n be a nonnegative irreducible symmetric matrix,and K = − K (cid:62) ∈ R n × n be a skew symmetric matrix, such that A = S + K ≥ . Let Q = ( Q (cid:62) ) − be an orthogonal matrix that diagonalizes S to Λ := Q (cid:62) SQ = r ( S ) 0 · · · λ · · · ... ... . . . ... · · · λ n . Then A has a constant Levinger’s function if and only if K := Q (cid:62) KQ = (cid:18) (cid:62) (cid:19) = (cid:0) (cid:1) ⊕ K , (16) where K = − K (cid:62) ∈ R n − × n − and (cid:62) = (0 . . . ∈ R n − .Proof. Since S is real and symmetric, S = QΛQ (cid:62) is in Jordan canonical form. Let x > be the normalized Perron vector of S . Then x = [ Q ] is the first column of Q ,and the other columns of Q are orthogonal to x , so x (cid:62) Q = (1 0 · · · A to have constant Levinger’s functionis that x (cid:62) K = (cid:62) , equivalent to x (cid:62) K = x (cid:62) QK Q (cid:62) = (1 0 · · · K Q (cid:62) = (cid:62) . Since Q is orthogonal, it has null space { } , so (1 0 · · · K Q (cid:62) = (cid:62) if and only if(1 0 · · · K = (cid:62) , which is the top row of K . K and K must be skew symmetricsince K is skew symmetric, as can be seen immediately from transposition. Theskew symmetry of K implies its first column must also be all zeros as its first rowis, establishing the form given in (16). (cid:3) Conclusions
We have shown that it is not in general true that the spectral radius along aline from a nonnegative square matrix A to its transpose — Levinger’s function — is concave. Our counterexamples to concavity have a simple principle in thecase of direct sums of block matrices, namely, that the maximum of two concavefunctions need not be concave. However, for the other examples we present —Toeplitz matrices, and positive circuit or cyclic weighted shift matrices — whateverprinciples underly the nonconcavity remain to be discerned. Also remaining to bediscerned are the properties of matrix families — a few of which we have presentedhere — that guarantee concave Levinger functions. A general characterization of therange of t for which the spectral radius is concave in Levinger’s homotopy remainsan open problem. Biographical Note
Bernard W. Levinger (Berlin, Germany, September 3, 1928 – Fort Collins, Col-orado, USA, January 17, 2020) and his family fled Nazi Germany to England in 1936,to Mexico in 1940, and to the United States in 1941, which initially placed them inan immigration prison and deported them to Mexico, but which ultimately allowedtheir immigration, whereupon they settled in New York City. Levinger graduatedfrom Bronx High School of Science and earned a doctorate in mathematics fromNew York University. He was Professor of Mathematics and Professor Emeritus atColorado State University, Fort Collins. He leaves a large family, including his wifeLory of more than 65 years.[15]
Acknowledgements
L.A. thanks Marcus W. Feldman for support from the Stanford Center for Com-putational, Evolutionary and Human Genomics and the Morrison Institute for Pop-ulation and Resources Studies, Stanford University; the Mathematical BiosciencesInstitute at The Ohio State University, for its support through U.S. National ScienceFoundation awards DMS-0931642 and DMS-1839810, “A Summit on New Interdis-ciplinary Research Directions on the Rules of Life”; and the Foundational QuestionsInstitute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley Commu-nity Foundation, for FQXi Grant number FQXi-RFP-IPW-1913. J.E.C. thanks theU.S. National Science Foundation for grant DMS-1225529 during the initial phase ofthis work and Roseanne K. Benjamin for help during this work.
References [1] Bapat, R.B., 1987. Two inequalities for the Perron root. Linear Algebra and ItsApplications 85, 241–248.[2] Bapat, R.B., Raghavan, T.E.S., 1997. Nonnegative Matrices and Applications.Cambridge University Press, Cambridge, UK. onconcavity of the Spectral Radius in Levinger’s Theorem 19 [3] B¨ottcher, A., Grudsky, S.M., 2005. Spectral Properties of Banded ToeplitzMatrices. Society for Industrial and Applied Mathematics, Philadelphia, PA.doi: .[4] Chien, M.T., Nakazato, H., 2020. Symmetry of cyclic weighted shift matriceswith pivot-reversible weights. The Electronic Journal of Linear Algebra 36,47–54.[5] Fiedler, M., 1995. Numerical range of matrices and Levinger’s theorem. LinearAlgebra and Its Applications 220, 171–180.[6] Hogben, L. (Ed.), 2014. Handbook of Linear Algebra. 2nd ed., Chapman andHall, Boca Raton, FL.[7] Horn, R.A., Johnson, C.R., 2013. Matrix Analysis. 2nd ed., Cambridge Univer-sity Press, Cambridge.[8] Levinger, B.W., 1970. An inequality for nonnegative matrices. Notices ofthe American Mathematical Society 17, 260. URL: .[9] Marek, I., 1974. An inequality involving positive kernels. ˇCasopis pro Pˇestov´an´ıMatematiky 99, 77–87.[10] Marek, I., 1978. Perron root of a convex combination of a positive kernel andits adjoint. Acta Universitatis Carolinae. Mathematica et Physica 19, 3–14.[11] Marek, I., 1984. Perron roots of a convex combination of a cone preserving mapand its adjoint. Linear Algebra and Its Applications 58, 185–200.[12] Psarrakos, P.J., Tsatsomeros, M.J., 2003. The Perron eigenspace of nonnegativealmost skew-symmetric matrices and Levinger’s transformation. Linear Algebraand Its Applications 360, 43–57.[13] Stanczak, S., Wiczanowski, M., Boche, H., 2009. Fundamentals of ResourceAllocation in Wireless Networks: Theory and Algorithms. volume 3. SpringerScience & Business Media.[14] Tsing, N.K., Fan, M.K., Verriest, E.I., 1994. On analyticity of functions involv-ing eigenvalues. Linear Algebra and Its Applications 207, 159–180.[15] Bernard Levinger, 1928–2020. URL: