Structured tools for structured matrices
TL;DR: In this paper, an extensive and unified collection of structure-preserving transformations for non-degenerate bilinear or sesquilinear forms on R n or C n is presented.
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Abstract: An extensive and unified collection of structure-preserving transformations is pre- sented and organized for easy reference. The structures involved arise in the context of a non- degenerate bilinear or sesquilinear form on R n or C n . A variety of transformations belonging to the automorphism groups of these forms, that imitate the action of Givens rotations, Householder reflec- tors, and Gauss transformations are constructed. Transformations for performing structured scaling actions are also described. The matrix groups considered in this paper are the complex orthogonal, real, complex and conjugate symplectic, real perplectic, real and complex pseudo-orthogonal, and pseudo-unitary groups. In addition to deriving new transformations, this paper collects and unifies existing structure-preserving tools.
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Fig. 4.3. Double Givens: complex symplectic direct sum embedding (left), symplectic concentric embedding (right). ![Table 3.1) there is no non-trivial scaling action by diagonal matrices on vectors in Cn. On the other hand, isotropic vectors in C2 can be arbitrary scaled by suitably chosen non-diagonal matrices in O(2,C). If x ∈ C2 is isotropic, then x is a complex scalar multiple of v = [1, i]T or w = [1,−i]T . Since diag(1,−1)w = v and diag(1,−1) ∈ O(2,C), we may assume, without loss of generality, that our isotropic vector is a](/figures/table-3-1-there-is-no-non-trivial-scaling-action-by-diagonal-3oq4l0hb.png)
Table 3.1) there is no non-trivial scaling action by diagonal matrices on vectors in Cn. On the other hand, isotropic vectors in C2 can be arbitrary scaled by suitably chosen non-diagonal matrices in O(2,C). If x ∈ C2 is isotropic, then x is a complex scalar multiple of v = [1, i]T or w = [1,−i]T . Since diag(1,−1)w = v and diag(1,−1) ∈ O(2,C), we may assume, without loss of generality, that our isotropic vector is a 
Fig. 4.4. Circle of α’s corresponding to pseudo-unitary G-reflectors that align x with ej . 
Fig. 4.2. Double Givens: symplectic direct sum embedding (left), symplectic concentric embedding (right). 
Fig. 4.5. Double Givens: conjugate symplectic direct sum embedding (left), symplectic concentric embedding (right). 
Fig. 4.1. Double Givens: perplectic direct sum embedding (left), perplectic interleaved embedding (right).
![Table 3.1) there is no non-trivial scaling action by diagonal matrices on vectors in Cn. On the other hand, isotropic vectors in C2 can be arbitrary scaled by suitably chosen non-diagonal matrices in O(2,C). If x ∈ C2 is isotropic, then x is a complex scalar multiple of v = [1, i]T or w = [1,−i]T . Since diag(1,−1)w = v and diag(1,−1) ∈ O(2,C), we may assume, without loss of generality, that our isotropic vector is a](/figures/table-3-1-there-is-no-non-trivial-scaling-action-by-diagonal-3oq4l0hb.webp)
Citations
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Beresford N. Parlett
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TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.
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Numerical Methods for Least Squares Problems
Åke Björck
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TL;DR: Theorems and statistical properties of least squares solutions are explained and basic numerical methods for solving least squares problems are described.
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The Symmetric Eigenvalue Problem.
TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.
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Lapack Users' Guide
Ed Anderson
- 01 Feb 1995
TL;DR: The third edition of LAPACK provided a guide to troubleshooting and installation of Routines, as well as providing examples of how to convert from LINPACK or EISPACK to BLAS.
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Numerical Linear Algebra
Lloyd N. Trefethen,David Bau +1 more
- 01 Jan 1997
Abstract: Exercise 1 (4 points) (a) Let unitary matrices Q 1 ,. .. , Q k ∈ C m×m be fixed and consider the problem of computing, for A ∈ C m×n , the product B = Q k · · · Q 1 A. Let the computation be carried out on a computer satisfying axioms (A1) and (A2). Show that this algorithm is backward stable. (Here A is thought of as data that can be perturbed; the matrices Q j are fixed and not to be perturbed.) (b) Give an example to show that this result no longer holds if the unitary matrices Q j are replaced with arbitrary matrices X j ∈ C m×m. Recall the axioms from exercise sheet 9, exercise 2: (A1) For all x ∈ R, there exists with || ≤ machine such that fl(x) = x(1 +). (A2) For all x, y ∈ F, there exists with || ≤ machine such that x y = (x * y)(1 +). Here * denotes an arithmetic operation and the floating point equivalent. Exercise 2 (4 points) Consider the example A = 1 1 1 1.0001 1 1.0001 , b = 2 0.0001 4.0001 . Do the computations in part (a) and (b) by hand. Show intermediate results where appropriate. Give exact results. From part (c) on, numerical results computed by MATLAB are acceptable. (a) What are the matrices A + and P = AA + for this example? Hint: Compute A + by solving the linear system (A * A)A + = A *. (b) Find the exact solutions x and y = Ax for the least squares problem Ax ≈ b. (c) What are κ(A), θ, and η? (d) What are the four relative condition numbers? (e) Give examples of perturbations δb and δA that approximately attain these four condition numbers.
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