aa r X i v : . [ m a t h . OA ] J a n DYNAMICAL DEFORMATION OF TOROIDAL MATRIXVARIETIES
FREDY VIDES
Abstract.
In this document we study the local connectivity of the sets whoseelements are m -tuples of pairwise commuting normal matrix contractions.Given ε >
0, we prove that there is δ > m -tuplesof pairwise commuting normal matrix contractions X := ( X , . . . , X m ) and˜ X := ( ˜ X , . . . , ˜ X m ) that are δ -close with respect to some suitable distance ð in ( C n × n ) m , we can find a m -tuple of matrix paths (homotopies) connecting X to ˜X relative to the intersection of some ε, ð -neighborhood of X with theset of m -tuples of pairwise commuting normal matrix contractions. One of thekey features of these matrix homotopies is that δ can be chosen independentof n .Some connections with topology and numerical matrix analysis will be out-lined as well. Introduction
In this document we study local deformation properties of what we call toroidalmatrix varieties. Given a m -tuple of n × n matrices X ∈ ( C n × n ) m and a suitabledistance ð in ( C n × n ) m induced by the spectral norm (that will be defined below),let us write N ð ( X , r ) to denote the set { Y ∈ ( C n × n ) m | ð ( X , Y ) ≤ r } . We call N ð ( X , r ) a r, ð -neighborhood of X .Given two matrices X, Y ∈ ( C n × n ), let us write X ∼ h Y to indicate that there isa homotopy between X and Y , in other words, there is function γ ∈ C ([0 , , C n × n )such that γ (0) = X and γ (1) = Y (where C ( X, Y ) denotes the algebra of continuousfunctions on X that take values in Y ). Given ε, δ >
0, any two m -tuples of pairwisecommuting normal matrix contractions X := ( X , . . . , X m ) and ˜ X := ( ˜ X , . . . , ˜ X m )such that ð ( X , ˜ X ) ≤ ε , let us write X ∼ h ˜ X to denote the homotopy induced by thehomotopies of their components. We study the existence of homotopies X ∼ h ˜ X relative to the intersection of some ε, ð -neighborhood of X with the set of m -tuplesof pairwise commuting normal matrix contractions.We can think of each homotopy X j ∼ h ˜ X j relative to N ð ( X j , ε ), as a noncommu-tative analogy of the family of curves ( links ) connecting the point sets determinedby the spectra σ ( X j ) and σ ( ˜ X j ) relative to the embedding of D × T in [ − , × T (where the embedding is induced by the mapping D → B (0 , ⊆ [ − , ). In a Date : May 30, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Matrix homotopy, matrix compression, normal matrix dilation, jointspectrum, pseudospectrum. similar way we can interpret the induced homotopies X ∼ h ˜ X as noncommuta-tive analogies of the links connecting the joint spectra σ ( X ) and σ ( ˜X ) relative to[ − , n × T .In this study we will use the C ∗ -algebraic technology developed in [10, 21, 22, 29]to derive a connectivity technique that can be used to study and solve problemsthat appear in clustered matrix approximation (in the sense of [28]) and matrixdilation/compression problems (in the sense of [12, 14] and [1, § § W ] described byL.2.1 in § T m ≃ T m of T m just a little bit , this analogy together with L.3.1 providesa connection with topologically controlled linear algebra (in the sense of [11]),which can be roughly described as the study of the relations between matrix setsand smooth manifolds that is performed by implementing techniques from geomet-ric topology in the study of matrix approximation problems. The search for thepreviously mentioned analogies and connections was motivated by a question raisedby M. H. Freedman regarding to the role played by the Kirby Torus Trick in linearalgebra and matrix approximation theory.In this document we build on the techniques developed in [22] and [29] to studythe analytic local connectivity properties of matrix representations of certain uni-versal C ∗ -algebras. The deformation technology constructed for this purpose can beadapted using the techniques presented in § § m -tuples in the senseof [23, 24], problems of this type appear in biomathematics, image processing andapplied spectral graph theory.Building on some of the ideas developed by M. A. Rieffel in [26, 27], by H. Lin in[21] and by P. Friis and M. R¨ordam in [10], we proved L.3.1, L.3.2 and L.3.3, theseresults together with L.2.1 provide us with the matrix approximation technologythat we use to prove the main results. The main results T.3.1, C.3.1 and T.3.2 arepresented in § § Preliminaries and Notation
Let (
X, d ) be a metric space. Then we say that ˜ X δ ⊂ X is a δ -dense if for all x ∈ X there exists ˜ x ∈ ˜ X such that d ( x, ˜ x ) ≤ δ . Given two compact subsets X, Y ofthe complex plane, we will write d H ( X, Y ) to denote the Hausdorff distance between
YNAMICAL DEFORMATION OF TOROIDAL MATRIX VARIETIES 3 X and Y that is defined by the expression d H ( X, Y ) := max { sup x ∈ X inf y ∈ Y | x − y | , sup y ∈ Y inf x ∈ X | y − x |} . Let us denote by ð the function ð : ( C n × n ) m × ( C n × n ) m → R +0 , ( S , T ) max j k S j − T j k . From here on k · k denotes the operator/spectralnorm. Given two topological objects, we use the expression X ≃ Y to denotea homeomorphism between them (a continuous function between the topologicalobjects that has a continuous inverse function). We will write D m and T m to denotethe m -dimensional closed unit disk and the m -dimensional torus respectively. Wewill write B ( x , r ) to denote the closed ball { x ∈ C | | x − x | ≤ r } .We will write M n to denote the set C n × n of n × n complex matrices, the sym-bols n and n will be used to denote the identity matrix and the zero matrix in M n respectively. Given a matrix A ∈ M n , we write A ∗ to denote the conjugatetranspose ¯ A ⊤ of A . A matrix X ∈ M n is said to be normal if XX ∗ = X ∗ X , amatrix H ∈ M n is said to be hermitian if H ∗ = H and a matrix U ∈ M n such that U ∗ U = U U ∗ = n is called unitary.A hermitian matrix P ∈ M n such that P = P is called a projector. Given twoprojectors P and Q , if P Q = QP = n we say that P and Q are orthogonal. By anorthogonal partition of unity in M n , we mean a finite set of orthogonal projectors { P j } in M n such that P j P j = n . We will omit the explicit reference to M n whenit is clear from the context. In this document by a matrix contraction we mean amatrix X in M n for some n ∈ Z + such that k X k ≤ X, Y ∈ M n we will write [ X, Y ] and Ad[ X ]( Y ) to denotethe operations [ X, Y ] := XY − Y X and Ad[ X ]( Y ) := XY X ∗ .Given a matrix A ∈ M n , we write σ ( X ) to denote the set { λ ∈ C | det( A − λ n ) =0 } of eigenvalues of A , the set σ ( A ) is called the spectrum of A . Given a compact set X ⊂ C and a subset S ⊆ M n , let us dote by S ( X ) the set S ( X ) := { X ∈ S | σ ( S ) ⊆ X } ,for instance, the expression N ( n )( D ) is used to denote the set normal contractionsin M n . Definition 2.1.
Given ε ≥ and a matrix X ∈ M n , we write σ ε ( X ) to denote the ε -Pseudospectrum of X which is the set defined by the following relations. σ ε ( X ) := { ˜ λ ∈ C | ˜ λ ∈ σ ( X + E ) , for some E ∈ M n with k E k ≤ ε } = { ˜ λ ∈ C | k X v − ˜ λ v k ≤ ε, for some v ∈ C n with k v k = 1 } Definition 2.2 (Semialgebraic Matrix Varieties) . Given J ∈ Z + , a system of J polynomials p , . . . , p J ∈ Π h N i = C h x , . . . , x N i in N NC-variables x , . . . , x N ∈ Π h N i and a real number ε ≥ , a particular matrix representation of the noncom-mutative semialgebraic set Z ε,n ( p , . . . , p J ) described by (2.1) Z ε,n ( p , . . . , p J ) := { X , . . . , X N ∈ M n | k p j ( X , . . . , X N ) k ≤ ε, ≤ j ≤ J } , will be called a ε, n -semialgebraic matrix variety ( ε, n -SMV), if ε = 0 we canrefer to the set as a matrix variety . Example 2.1.
Given any integer n ≥ , let us set N := diag [ n, n − , . . . , , wewill have that the set Z N := { X ∈ M n | [ N , X ] = 0 } is a matrix variety. If forsome δ > , we set now Z N ,δ := { X ∈ M n |k [ N , X ] k ≤ δ } , the set Z N ,δ is a matrixsemialgebraic variety. Example 2.2.
The subset of M n described by, (2.2) Z := (cid:26) ( X , X , X ) ∈ M n (cid:12)(cid:12)(cid:12)(cid:12) A X − X B = C ,A X − X B = C (cid:27) , FREDY VIDES where A , A , B , B , C and C are some fixed but arbitrary matrix contractionsin M n , is an algebraic matrix variety. Example 2.3.
Given ε > , the subset of M n described by, (2.3) Z := ( X , X , X ) ∈ M n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k A X − X B − C k ≤ ε, k A X − X B − C k ≤ ε,X j − X ∗ j = 0 , ≤ j ≤ m , where A , A , B , B , C and C are some fixed but arbitrary matrix contractionsin M n , is a semialgebraic matrix variety. Remark 2.1.
Matrix sets of the form 2.2 and 2.3 provide a connection betweenmatrix equations on words (in the sense of [17, 29] ) and algebraic/semialgebraic ma-trix varieties. From this perspective, the computation/refinement of matrix wordscorresponding to the numerical solution of matrix equations on words can be inter-preted as a discrete analogy of local matrix homotopies like the ones constructed inthe proofs of the results in § Definition 2.3 (Curved and Flat matrix paths) . Given any three hermitian ma-trices − n ≤ H , H , H ≤ n and a function f ∈ C ([ − , , and given any fournormal contractions D , . . . , D in M n , with D = Ad[ e πiH ]( D ) , D = f ( H ) and D = f ( H ) . Let us consider the paths ˘ Z ( t ) := Ad[ e πitH ]( D ) and ¯ V ( t ) := f ( tH + (1 − t ) H ) . We will say that ˘ Z is a curved interpolating path for D , D and we will say that the path ¯ V is a flat interpolating path for D , D . Definition 2.4 ( ⊛ operation) . Given two matrix paths
X, Y ∈ C ([0 , , M n ) wewrite X ⊛ Y to denote the concatenation of X and Y , which is the matrix pathdefined in terms of X and Y by the expression, X ⊛ Y s := (cid:26) X s , ≤ s ≤ ,Y s − , ≤ s ≤ . Definition 2.5 (Local matrix deformations x ε,S y ) . Given two matrices x, y ∈ M n we write x y if there is a matrix path z ∈ C ([0 , , M n ) such that z = x and z = y , if there is a ε -local matrix homotopy X ∈ C ([0 , , M n ) between x and y relative to the set S , we will write x ε,S y and will omit the explicit reference to S when it is clear from the context. It is often convenient to have N -tuples (or 2 N -tuples) of matrices with realspectra. For this purpose we use the following construction, initiated by McIntoshand Pryde. If X = ( X , . . . , X N ) is a N -tuple of n by n matrices then we canalways decompose X j in the form X j = X j + iX j where the X kj all have realspectra. We write π ( X ) := ( X , . . . , X N , X , . . . , X N ) and call π ( X ) a partitionof X . If the X kj all commute we say that π ( X ) is a commuting partition, and if the X kj are simultaneously triangularizable π ( X ) is a triangularizable partition. If the X kj are all semisimple (diagonalizable) then π ( X ) is called a semisimple partition.We say that N normal matrices X , . . . , X N ∈ M n are simultaneously diagonal-izable if there is a unitary matrix Q ∈ M n such that Q ∗ X j Q is diagonal for each j = 1 , . . . , N . In this case, for 1 ≤ k ≤ n , let Λ ( k ) ( X j ) := ( Q ∗ X j Q ) kk the ( k, k )element of Q ∗ X j Q , and set Λ ( k ) ( X , . . . , X N ) := (Λ ( k ) ( X ) , . . . , Λ ( k ) ( X N )) in C N .The set Λ( X , . . . , X N ) := { Λ ( k ) ( X , . . . , X N ) } ≤ k ≤ N YNAMICAL DEFORMATION OF TOROIDAL MATRIX VARIETIES 5 is called the joint spectrum of X , . . . , X N . We will write Λ( X j ) to denote the di-agonal matrix representation of the j -component of Λ( X , . . . , X N ), in other wordswe will have that Λ( X j ) = diag h Λ (1) ( X j ) , . . . , Λ ( n ) ( X j ) i . The following result ([22, L.4.1]) was proved in [22].
Lemma 2.1.
Given ε > there is δ = K m ε > such that, for any two N -tuples ofpairwise commuting normal matrices x = ( x , . . . , x N ) and y := ( y , . . . , y N ) suchthat ð ( x , y ) ≤ δ , there is a unitary matrix W such that [Ad[ W ]( x j ) , y j ] = 0 and max {k Ad[ W ]( x j ) − y j k , k Ad[ W ]( x j ) − x j k} ≤ ε , for each ≤ j ≤ N . Remark 2.2.
The constant K m in the statement of L.2.1 depends only m . Local Deformation of Normal Contractions
Applying functional calculus on commutative C ∗ -algebras we have that the jointspectrum σ ( X, Y ) of any two commuting hermitian matrix contractions
X, Y ∈ M n is contained in [ − , , moreover if k X + iY k ≤ σ ( X, Y ) ⊆ B (0 , ≃ D . Lemma 3.1.
Given r ∈ Z + and ν > , there is δ := ˆ δ ( ν, r ) > such that for anyunitary W and any normal contraction D in M n for n ≥ , if D = P rj =1 α j P j isdiagonal for ≤ r ∈ Z and α , . . . , α r ∈ D , the set { P j } is an orthogonal partitionof unity, and α j = α k whenever k = j , then there is a unitary matrix Z ∈ M n suchthat [ Z, D ] = 0 and k n − W Z k ≤ ν whenever k W DW ∗ − D k ≤ δ .Proof. Let µ := k W DW ∗ − D k = k W D − DW k . Then there are r continuousfunctions ℓ , . . . , ℓ r ∈ C ( D ) such that P k := ℓ k ( D ). By continuity one can find δ ( ν, r ) > k W P k W ∗ − P k k = k W ℓ k ( D ) W ∗ − ℓ k ( D ) k = k ℓ k ( W DW ∗ ) − ℓ k ( D ) k ≤ ν/ ( √ r ) < / ( √ r ), whenever µ ≤ δ . Since ν <
1, by perturbationtheory of C ∗ -algebras we will have that there is a unitary W j such that k − W j k ≤ √ k W P k W ∗ − P k k ≤ ν/r and W ∗ j P j W j = W P j W ∗ , and this implies that W j W P j = P j W j W for each j . Since { P j } is an orthogonal partition of unity, we willhave that ˜ W := P j W j W P j is a unitary matrix which satisfies the commutationrelation [ ˜ W , D ] = 0 together with the normed inequalities. k W − ˜ W k = k W X j P j − X j W j W P j k (3.1) = k X j (( n − W j ) W P j ) k (3.2) ≤ X j k n − W j k (3.3) ≤ X j r ν = r ( 1 r ν ) = ν (3.4)Let us set Z := ˜ W ∗ then k n − W Z k = k ˜ W − W k ≤ ν . This completes theproof. (cid:3) FREDY VIDES
Definition 3.1 (Pseudospectral retractive approximant) . Given δ > and anymatrix X ∈ M n , we say that the matrix ˜ X ∈ M n is a δ -Pseudospectral retractiveapproximant ( δ -PSRA ) of X if k X − ˜ X k ≤ δ and σ ( ˜ X ) is δ -dense in σ ( X ) . Lemma 3.2.
Given δ > and any hermitian matrix X ∈ M n such that k X k ≤ ,there is a hermitian δ -PSRA ˜ X δ of X such that [ X, ˜ X ] = n . Moreover, there are N δ distinct points { x , . . . , x N δ } and an orthogonal partition of unity { P , . . . , P N δ } such that ˜ X δ = P j x j P j .Proof. Let us suppose that n ≥ | σ ( X ) | ≥
2, as the proof is clear for scalar multiplesof n . Since σ ( X ) ⊆ [ − , M δ := 1 + ( δ ) − ∈ Z + . Then the finite set ˆ R δ ( X ) := { x k := − k − δ | ≤ k ≤ M δ } is δ -dense in[ − ,
1] with | R δ ( X ) | = M δ . Let us set S δ ( X ) := { ˇ x k := − k − δ | ≤ k ≤ N δ +1 } and P k := ˆ χ (ˇ x k , ˇ x k +1 ] ( X ) for each 1 ≤ k ≤ M δ , where ˆ χ ( c,d ] denotes a continuousrepresentation/approximation of the characteristic function χ ( c,d ] of ( c, d ] on someset S c,d such that ( c, d ] ⊆ S c,d ⊂ R . Then there is R δ ( X ) := { x j } ⊆ ˆ R δ ( X )that is δ -dense in σ ( X ) with x j ∈ (ˇ x k ( j ) , ˇ x k ( j )+1 ] for each x j ∈ R δ ( X ) and someˇ x k ( j ) , ˇ x k ( j )+1 ∈ S δ ( X ), and there is a set P δ ( X ) := { P , . . . , P N δ } ⊆ M n \{ n } thatis an orthogonal partition of unity such that [ P j , X ] = 0 and for each 1 ≤ k ≤ N δ there is x j ( k ) ∈ R δ ( X ) such that k XP k − x j ( k ) P k k ≤ δ with x k ( j ) = x k ( l ) whenever j = l . By the previous facts and spectral variation of normal matrices (in the senseof [5, 25]) we will have that if we set ˜ X δ := P k x j ( k ) P k with x j ( k ) ∈ R δ ( X ) for each k , then [ X, ˜ X δ ] = n and d H ( X, ˜ X δ ) ≤ k X − ˜ X δ k ≤ max k k XP k − x j ( k ) P k k ≤ δ .If necessary we can renumber the elements of R δ ( X ) according to the elements of P δ ( X ). This completes the proof. (cid:3) Remark 3.1.
We will refer to the finite sets R δ ( X ) and S δ ( X ) as representation and support grids respectively. The finite set P δ ( X ) will be called a δ -projective decomposition for X . Lemma 3.3.
Given δ > , m ≥ and any m -tuple of pairwise commuting hermit-ian matrix contractions X = ( X , . . . , X m ) ∈ M mn , there is a m -tuple of pairwisecommuting hermitian matrix contractions ˜ X δ = ( ˜ X δ, , . . . , ˜ X δ,m ) ∈ M mn such that [ X j , ˜ X δ,k ] = n and ˜ X δ,j is a δ -PSRA of X j for each ≤ j, k ≤ m . Moreover,there are N δ distinct points { x k, , . . . , x k,N δ } and an orthogonal partition of unity { P , . . . , P N δ } such that ˜ X δ,k = P j x k,j P j for each k .Proof. Since the joint spectrum σ ( X ) of X is a subset of [ − , m we can apply asimilar procedure to the one implemented in the proof of L.3.2 to find for each X j arepresentation grid ˆ R δ ( X j ) together with a support grid S δ ( X j ) and an associated δ -projective decomposition P δ ( X j ) such that | P δ ( X j ) | = N δ,j := | ˆ R δ ( X j ) | , wewill have that the set P δ ( X ) := { P , . . . , P N δ } = { P ,j P ,j · · · P m,j m | P k,j k ∈ P δ ( X k ) , ≤ j k ≤ N δ,k , ≤ k ≤ m } is an orthogonal partition of unity, by setting(1 + δ − ) m ≥ N δ := | P δ ( X ) | and ν := 1 / ( N δ − N δ ≥ N δ,k for each k , that ν ≤ δ and that there is a set R δ ( X j ) ⊆ { x j,k = 2( k − ν − | ≤ k ≤ N δ } of distinct points that is δ -dense in σ ( X j ) for each j and such that for each X k and each P j ∈ P δ ( ˆ X ), there is x k,j ∈ R δ ( X j ) such that k X k P j − x k,j P j k ≤ δ .Moreover, [ X j , P k ] = n for each 1 ≤ j ≤ m and each 1 ≤ k ≤ N δ . By a similarargument to the one implemented in the proof of L.3.2 it can be seen that the YNAMICAL DEFORMATION OF TOROIDAL MATRIX VARIETIES 7 matrix ˜ X δ,k := P j x k,j P j is a commuting hermitian δ -PSRA of X j for each j . Thiscompletes the proof. (cid:3) Definition 3.2 (Uniform piecewise analytic local connectivity in M mn ) . We willsay that a matrix variety Z m ⊆ M mn is uniformly locally piecewise analyticallyconnected ( ULPAC ) if given ε > , there is δ > such that for any two m -tuples X and Y in Z m such that ð ( X , Y ) ≤ δ ( X and Y are ( δ ε , ð ) -close), we have that X ∼ h Y relative to N ð ( X, ε ) ∩ Z m . Remark 3.2.
It is important to remark that ε and δ must not depend on n . Let us define the toroidal matrix varieties whose local connectivity will be studiedin this document.
Definition 3.3 ( I m ( n )) . The matrix variety I m ( n ) defined by I m ( n ) := ( X , . . . , X m ) ∈ M mn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ X j , X k ] = n X j − X ∗ j = n k X j k ≤ , ≤ j, k ≤ m will be called the m matrix cube . Definition 3.4 ( D m ( n )) . The matrix variety D m ( n ) defined by D m ( n ) := (cid:26) ( Z , . . . , Z m ) ∈ M mn (cid:12)(cid:12)(cid:12)(cid:12) [ Z j , Z k ] = [ Z j , Z ∗ j ] = n k Z j k ≤ , ≤ j, k ≤ m (cid:27) will be called the m matrix disk . Definition 3.5 ( T m ( n )) . The matrix variety T m ( n ) defined by T m ( n ) := (cid:26) ( U , . . . , U m ) ∈ M mn (cid:12)(cid:12)(cid:12)(cid:12) [ U j , U k ] = n U j U ∗ j = U ∗ j U j = n , ≤ j, k ≤ m (cid:27) will be called the m matrix torus . Remark 3.3.
It is important to notice that the components of two m -tuples X =( X , . . . , X m ) and Y = ( Y , . . . , Y m ) in I m ( n ) , D m ( n ) or T m ( n ) need not to satisfythe commutation relations [ X j , Y k ] = n for each j, k in general. Theorem 3.1.
The matrix variety I m ( n ) is uniformly locally piecewise analyticallyconnected.Proof. Without loss of generality we can assume that the components of Y are diag-onal matrices. As a consequence of L.2.1 we will have that given ν δ >
0, there is δ := K m ν δ > m -tuple X ∈ I m ( n ) that satisfies ð ( X , Y ) ≤ δ , thereexists a unitary matrix W such that [Ad[ W ]( X j ) , Y j ] = 0 and max {k Ad[ W ]( X j ) − Y j k , k Ad[ W ]( X j ) − X j k} ≤ ν δ , for each 1 ≤ j ≤ N . Since σ ( X j ) ⊆ [ − , ⊇ σ ( Y j )for each j . By joint spectral variation (in the sense of [25]) and by applying L.2.1again, we will have that W ∗ Λ( X j ) W = X j and k W ∗ Λ( X j ) W − Λ( X j ) k ≤ K m δ for each j . Then by L.3.3 there is a m -tuple of pairwise commuting hermit-ian δ -PSRA ˜ X δ of Λ ( X ) := (Λ( X ) , . . . , Λ( X m )), together with m representa-tion grids R δ ( X j ) := { x k, , . . . , x k,N δ } that are δ -dense in σ ( X j ) = σ (Λ( X j ))for each j , and a projective decomposition P δ ( Λ ( X )) := { P , . . . , P N δ } such that FREDY VIDES ˜ X δ,j = P j x k,j P j , [ X δ,j , Λ( X j )] = n and k X j − W ∗ ˜ X δ,j W k ≤ δ . Let us setˆ X δ := ( W ∗ X δ, W, . . . , W ∗ X δ,m W ). We will have that. k ˆ X δ,j − ˜ X δ,j k ≤ ð ( ˆ X δ , ˜ X δ )(3.5) ≤ ð ( ˆ X δ , X ) + ð ( X , Λ ( X )) + ð ( Λ ( X ) , ˜ X δ )(3.6) ≤ ( K m + 2) δ (3.7)Let us set N ε := | P δ ( Y ) | . If N ε = 1, there are four flat hermitian paths ¯ X ,j ( t ) := X j + t ( ˆ X δ,j − X j ), ¯ X ,j ( t ) := ˆ X δ,j + t ( ˜ X δ,j − ˆ X δ,j ), ¯ X ,j ( t ) := ˜ X δ,j + t (Λ( X j ) − ˜ X δ,j )and ¯ X ,j ( t ) := Λ( X j ) + t ( Y j − Λ( X j ) such that the matrix path ˆ X j := [( ¯ X ,j ⊛ ¯ X ,j ) ⊛ ¯ X ,j ] ⊛ ¯ X ,j ∈ C ([0 , , I m ( n )) solves the problem X j ε Y j for each j with ε = (2 K m + 3) δ and this implies that X ∼ h Y relative to N ð ( X , ε ) ∩ I m ( n ). If N ε ≥
2, then we can apply L3.1 to find a unitary ˆ W := ZW such that ˆ W ∗ ˜ X δ,j ˆ W = W ∗ ˜ X δ,j W = ˆ X δ,j and that k n − ˆ W k ≤ ε ( N ε , m, δ ) <
2, this implies that onecan find a hermitian matrix contraction ˆ K such that W = e iπ ˆ K . We have that thecurved path ˘ X ,j ( t ) := Ad[ e − iπ (1 − t ) ˆ K ]( ˜ X δ,j ) solves the problem ˆ X δ,j ε ˜ X δ,j foreach j with ε := 2 ε ( N ε , m, δ ). Let us set ˆ X j := [( ¯ X ,j ⊛ ˘ X ,j ) ⊛ ¯ X ,j ] ⊛ ¯ X ,j ∈ C ([0 , , I m ( n )) with ¯ X ,j , ¯ X ,j and ¯ X ,j defined as before. Then ˆ X j solves theproblem X j ε Y j for each j with ε := ε + ( K m + 1) δ , and this implies that X ∼ h Y relative to N ð ( X , ε ) ∩ I m ( n ). Let us set ε := max { ε , ε } . This completesthe proof. (cid:3) Corollary 3.1.
The matrix variety D m ( n ) is uniformly locally piecewise analyti-cally connected.Proof. Given any two m -tuples Z and S in D m ( n ), such that ð ( Z , S ) ≤ r/ r >
0, there are two semisimple commuting hermitian partitions π ( Z ) and π ( S ) of Z and S respectively, such that ð ( π ( Z ) , π ( S )) ≤ r . By the previously described factthe result follows by applying T.3.1 to the corresponding semisimple commutinghermitian partitions of any two ( δ ε , ð )-close m -tuples in D m ( n ). (cid:3) Theorem 3.2.
The matrix variety T m ( n ) is uniformly locally piecewise analyticallyconnected.Proof. We have that for any two unitary matrices
U, V ∈ M n such that [ U, V ] = n there are hermitian matrices H u , H v such that the flat path U ( t ) := e iπ ( H u + t ( H v − H u )) satisfies the interpolating conditions U (0) = U , U (1) = V , together with the con-straints [ U ( t ) , U ] = [ U ( t ) , V ] = n and U ( t ) U ( t ) ∗ = U ( t ) ∗ U ( t ) = n for each t ∈ [0 , U and any hermitian contraction K the curved path V ( t ) := Ad[ e iπtK ]( U ) preserves commutativity and is a unitary foreach t ∈ [0 , δ ε , ð )-close m -tuples in T m ( n ). (cid:3) Hints and Future Directions
The implications of the main results in § YNAMICAL DEFORMATION OF TOROIDAL MATRIX VARIETIES 9
The approximation and connectivity technology developed in this paper hasa natural connection to approximate simultaneous diagonalization of m -tuples ofmatrices (in the sense of [24]) and to normal matrix approximation of almost normalmatrices in the sense of [12, 16]. The development of numerical algorithms toperform these tasks will be the subject of future communications.The application and extension of the results in § § § Acknowledgement
I am very grateful with the Erwin Schr¨odinger Institute for Mathematical Physicsof the University of Vienna, for the outstanding hospitality and support during myvisit to participate in the research program on Topological phases of quantummatter in August of 2014. Much of the research reported in this document wascarried out while I was visiting the Institute.I am grateful with Terry Loring, Alexandru Chirvasitu, Moody Chu, Marc Ri-effel, Stan Steinberg, Jorge Destephen and Concepci´on Ferrufino, for several inter-esting questions and comments that have been very helpful for the preparation ofthis document.
ReferencesReferences [1]
K. M. R. Audenaert and F. Kittaneh.
Problems and Conjectures in Matrix and OperatorInequalities. arXiv:1201.5232v3 [math.FA] 2012.[2]
M. Ahues, A. Largillier, F. Dias d’Almeida and P. B. Vasconcelos.
Spectral refinementon quasi-diagonal matrices.
Linear Algebra Appl. 401 (2005) 109–117.[3]
M. Ahues, F. Dias d’Almeida, A. Largillier and P. B. Vasconcelos.
Spectral refine-ment for clustered eigenvalues of quasi-diagonal matrices.
Linear Algebra Appl. 413 (2006)394–402.[4]
D. Armentano and F. Cucker.
A Randomized Homotopy for the Hermitian EigenpairProblem.
Found Comput Math (2015) 15:281-312.[5]
R. Bhatia.
Matrix Analysis.
Gaduate Texts in Mathematics 169. Springer-Verlag. 1997.[6]
L. Chen, L. Han, and L. Zhou.
Computing Tensor Eigenvalues via Homotopy Methods.
SIAM J. Matrix Anal. Appl. Vol. 37, No. 1, pp. 290–319, 2016[7]
C. Davis, W. M. Kahan and H. F. Weinberger.
Norm-Preserving Dilations and TheirApplications to Optimal Error Bounds.
SIAM J. Numer. Anal. Vol. 19, No. 3, June 1982.[8]
M. T. Chu.
Linear Algebra Algorithms as Dynamical Systems.
Acta Numer. (2008), pp.001-086. 2008.[9]
J. E. Dennis, J. F. Traub and R. P. Weber.
The Algebraic Theory of Matrix Polynomials.
SIAM J. Numer. Anal. Vol. 13, No. 6, December 1976.[10]
P. Friis and M. R¨ordam.
Almost commuting self-adjoint matrices - a short proof of HuaxinLin’s theorem. J. Reine Angew. Math., 479:121–131, 1996.[11]
M. H. Freedman and W. H. Press.
Truncation of Wavelet Matrices: Edge Effects and theReduction of Topological Control
Linear Algebra Appl. 2:34:1-19 (1996)[12]
A. Greenbaum, T. Caldwell and K. Li.
Near Normal Dilations of Nonnormal Matricesand Linear Operators.
SIAM. J. Matrix Anal. Appl., Vol. 31, No. 4, pp. 1365-1381, 2016.[13]
C. J. Hillar and C. R. Johnson.
Symmetric Word Equations in two Positive DefiniteLetters.
Proceedings. Amer. Math. Soc., Volume 132, Number 4, Pages 945-953. S 0002-9939(03)07163-6 2003.[14]
J. Holbrook, N. Mudalige and R. Pereira.
Normal Matrix Compressions.
Oper. Matrices,Vol. 7, No. 4 (2013), 849-864. [15]
T.-M. Huang, W.-W. Lin and W. Wang.
A hybrid Jacobi–Davidson method for interiorcluster eigenvalues with large null-space in three dimensional lossless Drude dispersive metal-lic photonic crystals.
Comput. Phys. Comm. 207 (2016) 221-231.[16]
M. Huhtanen.
Aspects of nonnormality for iterative methods.
Linear Algebra Appl. 394(2005) 119-144.[17]
C. R. Johnson and C. Hillar.
Eigenvalues of Words in Two Positive De nite Letters.
SIAMJ. Matrix Anal. Appl., 23 (2002), 916-928. MR 2003e:81071[18]
Y.-F. Ke and C.-F. Ma.
Spectrum analysis of a more general augmentation block precon-ditioner for generalized saddle point matrices.
BIT Numer Math (2016) 56:489-500 DOI10.1007/s10543-015-0570-0.[19]
R. C. Kirby.
Stable homeomorphisms and the annulus conjecture.
Ann. Math., Second Series,Vol. 89, No. 3 (May, 1969), pp. 575-582[20]
Z.-Z. Liang and G.-F. Zhang.
Convergence behavior of generalized parameterized Uzawamethod for singular saddle-point problems.
J. Comput. Appl. Math. 311 (2017) 293–305.[21]
H. Lin.
Almost Commuting Selfadjoint Matrices and Applications.
In Operator algebras andtheir applications (Waterloo, ON, 1994/1995), volume 13 of Fields Inst. Commun., pages193-233. Amer. Math. Soc., Providence, RI, 1997.[22]
T. A. Loring and F. Vides.
Local Matrix Homotopies and Soft Tori. arXiv:1605.06590[math.OA]. 2016.[23]
T. Maehara and K. Murota.
Algorithm for Error-Controlled Simultaneous Block-Diagonalization of Matrices.
SIAM J. Matrix Anal. Appl., Vol. 32, No. 2, pp. 605-620, 2011.[24]
K. C. O’Meara and C. Vinsonhaler.
On approximately simultaneously diagonalizable ma-trices.
Linear Algebra Appl. 412 (2006) 39-74.[25]
A. J. Pryde. inequalities for the Joint Spectrum of Simultaneously Triangularizable Matri-ces.
Proc. Centre Math. Appl., Mathematical Sciences Institute, The Australian NationalUniversity (1992), 196-207.[26]
M. A. Rieffel.
Actions of Finite Groups on C*-Algebras.
Math. Scand. 47 (1980), 157-176.1980.[27]
M. A. Rieffel.
Vector Bundles and Gromov-Hausdorff Distance.
J. K-theory 5(2010), 39-103.[28]
B. Savas and I. S. Dhillon.
Clustered Matrix Approximation. SIAM J. Matrix Anal. Appl.Vol. 37, No. 4, pp. 1531-1555, 2016.[29]
F. Vides.
Local Deformation of Matrix Words. arXiv:1608.08562 [math.OA]
School of Mathematics and Computer Science, Department of Applied Mathemat-ics, Universidad Nacional Aut´onoma de Honduras, Ciudad Universitaria, Tegucigalpa,Honduras.
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