A class of normal dilation matrices affirming the Marcus-de Oliveira conjecture
aa r X i v : . [ m a t h . R A ] J un A CLASS OF NORMAL DILATION MATRICES AFFIRMING THEMARCUS-DE OLIVEIRA CONJECTURE
KIJTI RODTES
Abstract.
In this article, we prove a class of normal dilation matrices affirming theMarcus-de Oliveira conjecture.
Throughout, n will denote a positive integer. The determinant conjecture of Marcusand de Oliveira states that the determinant of the sum of two n by n normal matrices A and B belongs to the convex hull of the n ! σ -points, z σ := Q ni =1 ( a i + b σ ( i ) ), indexed by σ ∈ S n , where a i ’s and b j ’s are eigenvalues of A and B , respectively (see [9],[3],[11]). Webriefly write as ( A, B ) ∈ M OC if the pair of normal matrices
A, B affirms the Marcus andde Oliveira conjecture, i.e., det( A + B ) ∈ co ( { z σ | σ ∈ S n } ) . In [8], Fiedler showed that, for two hermitian matrices
A, B ∆( A, B ) := { det( A + U BU ∗ ) | U ∈ U n ( C ) } is a line segment with σ -points as endpoints, where U n ( C ) denotes the set of all uni-tary matrices of dimension n × n . This result, in fact, motivates the conjecture. As aconsequence of Fiedler’s result, ( A, B ) ∈ M OC for any pair of skew-hermitian matrices
A, B .In [1], N. Bebiano, A. Kovacec, and J.da Providencia provided that if A is positivedefinite and B a non-real scalar multiple of a hermitian matrix, then ( A, B ) ∈ M OC .They also obtained that if eigenvalues of A are pairwise distinct complex numbers lyingon a line l and all eigenvalues of B lie on a parallel to l , then ( A, B ) ∈ M OC . S.W.Drury showed that (
A, B ) ∈ M OC for the case that A is hermitian and B is non-realscalar multiple of a hermitian matrix (essentially hermitian matrix) in [5] and the casethat A = sU and B = tV for s, t ∈ C and U, V ∈ U n ( C ) in [6].It is also known that, for normal matrices A, B ∈ M n ( C ) (the set of all n × n matricesover C ), ( A, B ) ∈ M OC : if det( A + B ) = 0 ([7]); if the point z σ lie all on a straight line([10]); if n = 2 , A or B has only two distinct eigenvalues, one of them simple, Keywords: Normal dilation, Normal matrices, Marcus de Oliveira ConjectureMSC(2010): 15A15; 15A60; 15A86 . ([3]). However, it seems that there is no new affirmative class of normal matrices to thisconjecture after the year 2007.Let X be a square n × n complex matrix and s be a complex number. It is a directcalculation to see that N ( X, s ) := (cid:18) X ( X − sI ) ∗ ( X − sI ) ∗ X (cid:19) is a normal matrix of size 2 n × n and thus it is a normal dilation of X . We will see (in theproof of the main result) that the eigenvalues of N ( X, s ) lie on both real and imaginaryaxis and thus this matrix need not be essentially hermitian or a scalar multiple of a unitarymatrix. In this short note, we show that:
Theorem 0.1.
Let
X, Y ∈ M n ( C ) and s, t ∈ C . Then ( N ( X, s ) , N ( Y, t )) ∈ M OC . Note that if A ∈ M n ( C ) is normal then U AU ∗ is also normal for any U ∈ U n ( C ).Then V N ( X, s ) V ∗ is also a normal dilation of X for any V ∈ U n ( C ). Moreover, sincethe conjecture is invariant under simultaneous unitary similarity, we also deduce fromTheorem 0.1 that ( V N ( X, s ) V ∗ , V N ( Y, t ) V ∗ ) ∈ M OC for any V ∈ U n ( C ).To prove the main result, we will use the following lemmas. Lemma 0.2.
Let
A, B ∈ M n ( C ) and C, D ∈ M m ( C ) be normal. If ( A, B ) ∈ M OC and ( C, D ) ∈ M OC , then ( A ⊕ C, B ⊕ D ) ∈ M OC .Proof.
Suppose that { a i | ≤ i ≤ n } , { b i | ≤ i ≤ n } , { c i | ≤ i ≤ m } and { d i | ≤ i ≤ m } are ordered set of the eigenvalues of A, B, C and D , respectively. Denote e i := a i , f i := b i for i = 1 , . . . , n and e n + j = c j , f n + j = d j for j = 1 , . . . , m . Then, { e i | ≤ i ≤ n + m } and { f i | ≤ i ≤ n + m } are ordered set of the eigenvalues of A ⊕ C and B ⊕ D , respectively.For each σ ∈ S n , π ∈ S m and θ ∈ S n + m , denote z σ , v π and w θ the product Q ni =1 ( a i + b σ ( i ) ), Q mi =1 ( c i + d π ( i ) ) and Q n + mi =1 ( e i + f θ ( i ) ), respectively. Suppose that ( A, B ) ∈ M OC and(
C, D ) ∈ M OC , thendet( A + B ) = X σ ∈ S n t σ z σ and det( C + D ) = X π ∈ S m s π v π , where t σ , s π ∈ [0 ,
1] such that P σ ∈ S n t σ = 1 and P σ ∈ S m s π = 1. Note thatdet( A ⊕ C + B ⊕ D ) = det(( A + B ) ⊕ ( C + D ))= det( A + B ) · det( C + D )= ( X σ ∈ S n t σ z σ )( X π ∈ S m s π v π )= X σ ∈ S n ,π ∈ S m ( t σ s π )( z σ v π ) . For each σ ∈ S n and π ∈ S m , define a permutation θ ( σ, π ) ∈ S n + m by θ ( σ, π ) := (cid:18) · · · n n + 1 · · · n + mσ (1) · · · σ ( n ) n + π (1) · · · n + π ( m ) (cid:19) Then w θ ( σ,π ) = z σ v π . Since, for each σ ∈ S n and π ∈ S m , t σ s π ∈ [0 ,
1] and X σ ∈ S n ,π ∈ S m ( t σ s π ) = ( X σ ∈ S n t σ )( X σ ∈ S m s π ) = (1)(1) = 1 , we conclude thatdet( A ⊕ C + B ⊕ D ) ∈ co { w θ ( σ,π ) | σ ∈ S n , π ∈ S m } ⊆ co { w θ | θ ∈ S n + m } . Hence ( A ⊕ C, B ⊕ D ) ∈ M OC . (cid:3) To be a self contained material, we record a result of S.W. Drury.
Theorem 0.3. [4]
Let A and B be hermitian matrices with the given eigenvalues ( a , . . . , a n ) and ( b , . . . , b n ) respectively. Let ( t , . . . , t n ) be the eigenvalues of A + B . Then n Y j =1 ( λ + t j ) ∈ co { n Y j =1 ( λ + a j + b σ ( j ) ) | σ ∈ S n } , where co denotes the convex hull in the space of polynomials and λ is an indeterminate. As a corollary of the above theorem, we have that:
Lemma 0.4.
Let
X, Y ∈ M n ( C ) and α, β ∈ C . Then ( X − X ∗ + αI n , Y − Y ∗ + βI n ) ∈ M OC and ( X + X ∗ + αI n , Y + Y ∗ + βI n ) ∈ M OC .Proof.
Since X + X ∗ and Y + Y ∗ are hermitian, by Theorem 0.3, we deduce directly that( X + X ∗ + αI n , Y + Y ∗ + βI n ) ∈ M OC . Since X − X ∗ and Y − Y ∗ are skew-hermitian, i ( X − X ∗ ) and i ( Y − Y ∗ ) are hermitian. Again, by Theorem 0.3, ( X − X ∗ + αI n , Y − Y ∗ + βI n ) ∈ M OC . (cid:3) Proof. (Theorem ) Let U be the block matrix in M n ( C ) defined by U := 1 √ (cid:18) I n I n − I n I n (cid:19) . It is a direct computation to see that U is a unitary matrix and U ∗ (cid:18) M NN M (cid:19) U = ( M − N ) ⊕ ( M + N ) , for any M, N ∈ M n ( C ). Let A := X − X ∗ + ( s ) I n , B := Y − Y ∗ + tI n , C := X + X ∗ − ( s ) I n ,and D := Y + Y ∗ − tI n . By Lemma 0.4, the pair of normal matrices ( A, B ) and (
C, D )satisfy the conjecture. Hence, by Lemma 0.2, ( A ⊕ C, B ⊕ D ) ∈ M OC . Therefore,( N ( X, s ) , N ( Y, t )) = ( U ( A ⊕ C ) U ∗ , U ( B ⊕ D ) U ∗ ) ∈ M OC, which completes the proof. (cid:3)
Acknowledgments
The author would like to thank Prof Tin Yau Tam for bringing this topic to the author.He would like to thank the referee(s) for valuable comments to improve the paper. He alsowould like to thank Naresuan University for the financial support on the project numberR2563C006.
Kijti Rodtes
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