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4. For any n x n matrix A there exists an n x n unitary matrix (U* = U-¹) such that U* AU = T (1) where T is an n x n matrix in upper triangular form. Equation (1) is called a Schur decomposition. The diagonal elements of T are the eigenvalues of A. Note that such a decomposition is not unique. An iterative algorithm to find a Schur decomposition for an n x n matrix is as follows. It generates at each step matrices Uk and Tk (k = 1,...,n-1) with the proper- ties: each Uk is unitary, and each Tk has only zeros below its main diagonal in its first k columns. Tn-1 is in upper triangular form, and U = U₁U₂... Un-1 is the unitary matrix that transforms A into Tn-1. We set To= A. The k-th step in the iteration is as follows. Step 1. Denote as Ak the (n − k + 1) × (n − k + 1) submatrix in the lower right portion of Tk-1. Step 2. Determine an eigenvalue and the corresponding normalized eigenvector for Ak. Step 3. Construct a unitary matrix N which has as its first column the nor- malized eigenvector found in step 2. Step 4. For k = 1, set U₁ = N₁, for k > 1, set 0 Uk= (Ik-1 N₂) 0 where Ik-1 is the (k − 1) × (k − 1) identity matrix. Step 5. Calculate Tk = UT-1Uk. Apply the algorithm to the 3 x 3 symmetric matrix A = - (¦ ¦ ;) 101 010 101

4. For any n x n matrix A there exists an n x n unitary matrix (U* = U-¹) such that U* AU = T (1) where T is an n x n matrix in upper triangular form. Equation (1) is called a Schur decomposition. The diagonal elements of T are the eigenvalues of A. Note that such a decomposition is not unique. An iterative algorithm to find a Schur decomposition for an n x n matrix is as follows. It generates at each step matrices Uk and Tk (k = 1,...,n-1) with the proper- ties: each Uk is unitary, and each Tk has only zeros below its main diagonal in its first k columns. Tn-1 is in upper triangular form, and U = U₁U₂... Un-1 is the unitary matrix that transforms A into Tn-1. We set To= A. The k-th step in the iteration is as follows. Step 1. Denote as Ak the (n − k + 1) × (n − k + 1) submatrix in the lower right portion of Tk-1. Step 2. Determine an eigenvalue and the corresponding normalized eigenvector for Ak. Step 3. Construct a unitary matrix N which has as its first column the nor- malized eigenvector found in step 2. Step 4. For k = 1, set U₁ = N₁, for k > 1, set 0 Uk= (Ik-1 N₂) 0 where Ik-1 is the (k − 1) × (k − 1) identity matrix. Step 5. Calculate Tk = UT-1Uk. Apply the algorithm to the 3 x 3 symmetric matrix A = - (¦ ¦ ;) 101 010 101

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.2: Diagonalization
Problem 32E
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4. For any n x n matrix A there exists an n x n unitary matrix (U* = U-¹) such that U* AU = T (1) where T is an n x n matrix in upper triangular form. Equation (1) is called a Schur decomposition. The diagonal elements of T are the eigenvalues of A. Note that such a decomposition is not unique. An iterative algorithm to find a Schur decomposition for an n x n matrix is as follows. It generates at each step matrices Uk and Tk (k = 1,...,n-1) with the proper- ties: each Uk is unitary, and each T has only zeros below its main diagonal in its first k columns. Tn-1 is in upper triangular form, and U = U₁U₂... Un-1 is the unitary matrix that transforms A into Tn-1. We set To A. The k-th step in the iteration is as follows. Step 1. Denote as Ak the (n − k + 1) × (n − k + 1) submatrix in the lower right portion of Tk-1. Step 2. Determine an eigenvalue and the corresponding normalized eigenvector for Ak. Step 3. Construct a unitary matrix Nk which has as its first column the nor- malized eigenvector found in step 2. Step 4. For k = 1, set U₁ = N₁, for k > 1, set Ik-1 0 Uk = = (¹ - (16-1 N₂) 0 where Ik-1 is the (k − 1) × (k − 1) identity matrix. Step 5. Calculate Tk = UT-1Uk. Apply the algorithm to the 3 x 3 symmetric matrix 101 A = 010 1 0 1
Transcribed Image Text:4. For any n x n matrix A there exists an n x n unitary matrix (U* = U-¹) such that U* AU = T (1) where T is an n x n matrix in upper triangular form. Equation (1) is called a Schur decomposition. The diagonal elements of T are the eigenvalues of A. Note that such a decomposition is not unique. An iterative algorithm to find a Schur decomposition for an n x n matrix is as follows. It generates at each step matrices Uk and Tk (k = 1,...,n-1) with the proper- ties: each Uk is unitary, and each T has only zeros below its main diagonal in its first k columns. Tn-1 is in upper triangular form, and U = U₁U₂... Un-1 is the unitary matrix that transforms A into Tn-1. We set To A. The k-th step in the iteration is as follows. Step 1. Denote as Ak the (n − k + 1) × (n − k + 1) submatrix in the lower right portion of Tk-1. Step 2. Determine an eigenvalue and the corresponding normalized eigenvector for Ak. Step 3. Construct a unitary matrix Nk which has as its first column the nor- malized eigenvector found in step 2. Step 4. For k = 1, set U₁ = N₁, for k > 1, set Ik-1 0 Uk = = (¹ - (16-1 N₂) 0 where Ik-1 is the (k − 1) × (k − 1) identity matrix. Step 5. Calculate Tk = UT-1Uk. Apply the algorithm to the 3 x 3 symmetric matrix 101 A = 010 1 0 1
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