On tridiagonal matrices unitary equivalent with normal matrices Raf Vandebril Departement of Computer Science K.U.Leuven Cortona 2008
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Contents Unitary Equivalence relation 1 Householder equivalence tridiagonalization Lanczos equivalence tridiagonalization Essential uniqueness The normal case 2 Main theorem Scalar product spaces Specific reductions Few extra properties Associated Krylov spaces 3 Krylov subspaces Krylov matrices Examples Eigenvalues and singular values 4 5 Conclusions On tridiagonal matrices unitary equivalent, with normal matrices 2 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Outline Unitary Equivalence relation 1 Householder equivalence tridiagonalization Lanczos equivalence tridiagonalization Essential uniqueness The normal case 2 Main theorem Scalar product spaces Specific reductions Few extra properties Associated Krylov spaces 3 Krylov subspaces Krylov matrices Examples Eigenvalues and singular values 4 5 Conclusions On tridiagonal matrices unitary equivalent, with normal matrices 3 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Householder equivalence tridiagonalization Given A ∈ C n × n , U k and V k Householder transformations: U H V H k x = ω � x � e 1 , | ω | = 1 , and k y = σ � y � e 1 , | σ | = 1 . Algorithm (Householder equivalence tridiagonalization) The algorithm computes U H AV = T, with T tridiagonal, U and V unitary. For k=1:n-2 Compute the Householder reflector U k = I − α vv H , based on A ( k + 1 : n , k ) A ( k + 1 : n , k : n ) = U H k A ( k + 1 : n , k : n ) Compute the Householder reflector V k = I − β ww H , based on A ( k , k + 1 : n ) H A ( k : n , k + 1 : n ) = A ( k : n , k + 1 : n ) V k end On tridiagonal matrices unitary equivalent, with normal matrices 4 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Lanczos equivalence tridiagonalization Suppose U H AV = T , having diagonal elements α i , subdiagonals β i and superdiagonals γ i and U = [ u 1 ,..., u n ] and V = [ v 1 ,..., v n ] . Based on A H U = VT H AV = UT and we get A v k = γ k − 1 u k − 1 + α k u k + β k u k + 1 (1) A H u k = β k − 1 v k − 1 + α k v k + γ k v k + 1 , (2) Rewriting (1) and (2) gives us (with α k = u H k A v k = v H k A H u k ): r k + 1 = A v k − γ k − 1 u k − 1 − α k u k , A H u k − β k − 1 v k − 1 − α k v k . s k + 1 = Hence β k = � r k + 1 � 2 , u k + 1 = r k + 1 / β k and γ k = � s k + 1 � 2 , v k + 1 = s k + 1 / γ k . On tridiagonal matrices unitary equivalent, with normal matrices 5 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Lanczos equivalence Tridiagonalization Algorithm (Lanczos equivalence tridiagonalization) The algorithm computes “theoretically” U H AV = T, with T tridiagonal, U and V unitary. Initialize u 1 and v 1 . (E.g., u 1 = e 1 = v 1 .) for k = 1 : n − 1 α k = u H k A v k r = A v k − γ k − 1 u k − 1 − α k u k s = A H u k − β k − 1 v k − 1 − α k v k β k = ω � r � 2 , γ k = σ � s � 2 ( ω , σ are free, | ω | = | σ | = 1 ) u k + 1 = r / β k , v k + 1 = s / γ k end On tridiagonal matrices unitary equivalent, with normal matrices 6 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Essential uniqueness: Case 1 Case 1: sub- and superdiagonal elements different from zero. Theorem A ∈ C n × n , U , V unitary, T , S tridiagonal: S = ˆ U H A ˆ T = U H AV , V . sub- and superdiagonal elements different from zero. When ω ˆ V e 1 = ω ˆ U e 1 = ˆ U e 1 , V e 2 , | ω 1 | = | ˆ ω 1 | = 1 . then unitary diagonal D and ˆ D exist, such that VD = ˆ U ˆ D = ˆ V , U and | T | = | S | . On tridiagonal matrices unitary equivalent, with normal matrices 7 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Essential uniqueness: Case 2 Case 2: sub- and superdiagonal elements can be zero. Theorem Same assumptions as before; K = min { i | s i + 1 , i = 0 } , and L = min { i | s i , i + 1 = 0 } . Then we have three different cases: K < L. Columns 1 up to K of U and ˆ U are essentially unique. Columns 1 up to K + 1 of V and ˆ V are essentially unique. For 1 ≤ k ≤ K and 1 ≤ l ≤ K + 1 : | t k , l | = | s k , l | . L < K. Similar. K = L. Similar. On tridiagonal matrices unitary equivalent, with normal matrices 8 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Essential uniqueness: Case 2 Below the resulting T is depicted: The ⊠ denote the essentially unique parts. K < L and K = 3 K > L and L = 3 K = L = 3 ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ ⊠ 0 ⊠ ⊠ 0 0 × × ⊠ × × 0 × × × × × × × × On tridiagonal matrices unitary equivalent, with normal matrices 9 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Outline Unitary Equivalence relation 1 Householder equivalence tridiagonalization Lanczos equivalence tridiagonalization Essential uniqueness The normal case 2 Main theorem Scalar product spaces Specific reductions Few extra properties Associated Krylov spaces 3 Krylov subspaces Krylov matrices Examples Eigenvalues and singular values 4 5 Conclusions On tridiagonal matrices unitary equivalent, with normal matrices 10 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Main theorem Theorem Given a normal A ∈ C n × n . For U , V, with U e 1 = ω V e 1 ( | ω | = 1 ) such that U H AV = T with T tridiagonal having subdiagonal elements β i , superdiagonal elements γ i . We have (assume γ i and β i different from 0 ): | β i | = | γ i | , ∀ i = 1 ,..., n − 1 . In case a γ i and/or β i is zero, a sort of restart or equivalently and extra relation needs to be put on U and V . On tridiagonal matrices unitary equivalent, with normal matrices 11 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Comments on the proof By induction on k (three steps): | γ k | = | β k | . 1 A recurrence in bivariate polynomials is proven for A H u k + 1 and A v k + 1 : 2 � � 1 A H β 1 : k − 1 A H u k + 1 p k ( A H , A ) − β k − 1 γ k − 1 p k − 1 ( A , A H ) − α k p k ( A , A H ) = v 1 β 1 : k γ 1 : k − 1 1 p k + 1 ( A , A H ) v 1 = β 1 : k and a similar relation 1 p k + 1 ( A H , A ) v 1 , A v k + 1 = γ 1 : k β 0 = γ 0 = 0, p 0 = 0 and p 1 ( x , y ) = y . Based on these results we get � A v k + 1 � 2 = � A H u k + 1 � 2 . 3 This has also consequences on the implementation. On tridiagonal matrices unitary equivalent, with normal matrices 12 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Scalar product spaces For A normal we have a factorization U H AV = T = SD , with S complex symmetric and D unitary diagonal. On tridiagonal matrices unitary equivalent, with normal matrices 13 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Scalar product spaces For A normal we have a factorization U H AV = T = SD , with S complex symmetric and D unitary diagonal. Consider the bilinear form ( Ω is a weight matrix): � x , y � Ω = x T Ω y . The adjoint of A w.r.t. �· , ·� Ω is A ⋆ : � A x , y � Ω = � x , A ⋆ y � Ω , x , y ∈ C n . for A closed formula: A ⋆ = Ω − 1 A T Ω , On tridiagonal matrices unitary equivalent, with normal matrices 13 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Scalar product spaces For A normal we have a factorization U H AV = T = SD , with S complex symmetric and D unitary diagonal. Consider the bilinear form ( Ω is a weight matrix): � x , y � Ω = x T Ω y . The adjoint of A w.r.t. �· , ·� Ω is A ⋆ : � A x , y � Ω = � x , A ⋆ y � Ω , x , y ∈ C n . for A closed formula: A ⋆ = Ω − 1 A T Ω , It is easily checked that for Ω = D : D − 1 T T D , T ⋆ = D − 1 ( SD ) T D , = = T . Hence, T is self-adjoint w.r.t. �· , ·� D . On tridiagonal matrices unitary equivalent, with normal matrices 13 / 34
Unitary Equivalence relation The normal case Associated Krylov spaces Eigenvalues and singular values Conclusions Some corollaries Compact formulation of the main theorem. Theorem For A ∈ C n × n normal and U H AV = T, satisfying the conditions above we get: T is self-adjoint w.r.t. �· , ·� Ω , with Ω a unitary diagonal matrix. On tridiagonal matrices unitary equivalent, with normal matrices 14 / 34
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