Takagi Factorization Research Articles

Riemannian optimization methods for the truncated Takagi factorization

Riemannian optimization methods for the truncated Takagi factorization

Fifty Three Matrix Factorizations: A Systematic Approach

The success of matrix factorizations such as the singular value decomposition (SVD) has motivated the search for even more factorizations. We catalog 53 matrix factorizations, most of which we believe to be new. Our systematic approach, inspired by the generalized Cartan decomposition of Lie theory, also encompasses known factorizations such as the SVD, the symmetric eigendecomposition, the CS decomposition, the hyperbolic SVD, structured SVDs, the Takagi factorization, and others thereby covering familiar matrix factorizations as well as ones that were waiting to be discovered. We suggest that Lie theory has one way or another been lurking hidden in the foundations of the very successful field of matrix computations with applications routinely used in so many areas of computation. In this paper, we investigate consequences of the Cartan decomposition and the little known generalized Cartan decomposition for matrix factorizations. We believe that these factorizations once properly identified can lead to further work on algorithmic computations and applications.

Takagi Factorization of Matrices Depending on Parameters and Locating Degeneracies of Singular Values

In this work we consider the Takagi factorization of a matrix valued function depending on parameters. We give smoothness and genericity results and pay particular attention to the concerns caused by having either a singular value equal to $0$ or multiple singular values. For these phenomena, we give theoretical results showing that their co-dimension is $2$, and we further develop and test numerical methods to locate in parameter space values where these occurrences take place. Numerical study of the density of these occurrences is performed.

Takagi topological insulator with odd PT pairs of corner states

We present a novel class of topological insulators, termed the Takagi topological insulators (TTIs), which is protected by the sublattice symmetry and spacetime inversion ($\mathcal P\mathcal T$) symmetry. The required symmetries for the TTIs can be realized on any bipartite lattice where the inversion exchanges sublattices. The protecting symmetries lead to the classifying space of Hamiltonians being unitary symmetric matrices, and therefore Takagi's factorization can be performed. Particularly, the global Takagi's factorization can (cannot) be done on a $3$D ($2$D) sphere. In 3D, there is a $\mathbb{Z}_2$ topological invariant corresponding to the parity of the winding number of Takagi's unitary-matrix factor over the entire Brillouin zone, where the $\mathbb Z_2$ nature comes from the $O(N)$ gauge degrees of freedom in Takagi's factorization. In 2D, the obstruction for a global Takagi's factorization is characterized by another $\mathbb{Z}_2$ topological invariant, equivalent to the second Stiefel-Whitney number. For the third-order topological phases, the $3$D TTIs feature a parity condition for corner zero-modes, i.e., there always exist odd $\mathcal P\mathcal T$ pairs of corners with zero-modes. Moreover, for any $\mathcal P\mathcal T$ invariant sample geometry, all configurations of corner zero-modes satisfying the parity condition can exist with the same nontrivial bulk topological invariant. Actually, without closing the bulk gap, the boundary phase diagram have a cellular structure, where each topological boundary phase associated with a particular (cross-order) boundary-mode pattern corresponds to a contractible cell with certain dimension in the parameter space.

On the two-sided perplectic singular value decomposition and perplectically diagonalizable matrices

Necessary and sufficient conditions for a square matrix [Formula: see text] to be written as [Formula: see text], where [Formula: see text] is a diagonal matrix, and [Formula: see text] and [Formula: see text] are isometries of either the Minkowski indefinite inner product or the Symplectic scalar product, are known. We give the analogue of the preceding decomposition to the perplectic scalar product [Formula: see text], where [Formula: see text] is the backward identity matrix. We also give a perplectic analogue of the Takagi factorization for symmetric matrices. As a related result, we give necessary and sufficient conditions for a perplectic matrix to be diagonalizable by a perplectic matrix, and we show that every perplectic matrix is a product of perplectic matrices which are diagonalizable by a perplectic matrix.

A tale of three diagonalizations

In addition to the diagonalization of a normal matrix by a unitary similarity transformation, there are two other types of diagonalization procedures that sometimes arise in quantum theory applications — the singular value decomposition and the Autonne–Takagi factorization. In this pedagogical review, each of these diagonalization procedures is performed for the most general [Formula: see text] matrices for which the corresponding diagonalization is possible, and explicit analytical results are provided in each of the three cases.

Generalized Pauli–Gursey transformation and Majorana neutrinos

We discuss a generalization of the Pauli–Gursey transformation, which is motivated by the Autonne–Takagi factorization, to an arbitrary n number of generations of neutrinos using U(2n) that defines general canonical transformations and diagonalizes symmetric complex Majorana mass matrices in special cases. The Pauli–Gursey transformation mixes particles and antiparticles and thus changes the definition of the vacuum and C. We define C, P and CP symmetries at each Pauli frame specified by a generalized Pauli–Gursey transformation. The Majorana neutrinos in the C and P violating seesaw model are then naturally defined by a suitable choice of the Pauli frame, where only Dirac-type fermions appear with well-defined C, P and CP, and thus the C symmetry for Majorana neutrinos agrees with the C symmetry for Dirac-type fermions. This fully symmetric setting corresponds to the idea of Majorana neutrinos as Bogoliubov quasi-particles. In contrast, the conventional direct construction of Majorana neutrinos in the seesaw model, where CP is well-defined but C and P are violated, encounters the mismatch of C symmetry for Majorana neutrinos and C symmetry for chiral fermions; this mismatch is recognized as the inevitable appearance of the singlet (trivial) representation of C symmetry for chiral fermions.

Partial orthogonal rank-one decomposition of complex symmetric tensors based on the Takagi factorization

This paper is devoted to the computation of rank-one decomposition of complex symmetric tensors. Based on the Takagi factorization of complex symmetric matrices, we derive algorithm for computing the partial orthogonal rank-one decomposition of complex symmetric tensors with an order being a power of two, denoted by CSTPOROD . We consider the properties of this decomposition. We design a strategy (tensor embedding) to computing the partial orthogonal rank-one decomposition of complex symmetric tensors, whose order is not the power of two. Similar to the case of complex symmetric tensors, we consider how to compute the partial orthogonal rank-one decomposition of general complex tensors. We illustrate our algorithms via numerical examples.

Iterative algorithms for computing US- and U-eigenpairs of complex tensors

This paper is devoted to the computation of US-eigenpairs of complex symmetric tensors and U-eigenpairs of complex tensors. Based on the Takagi factorization of complex symmetric matrices, we derive an iterative algorithm for computing US-eigenpairs of complex symmetric tensors, denoted as QRCST Algorithm. We also observe that multiple US-eigenpairs can be found from a local permutation heuristic, which is effectively a tensor similarity transformation, resulting in the permuted version of QRCST. We then generalize our techniques to general complex tensors. Finally, we derive a higher order power type method for computing a US- or a U-eigenpair, similar to the higher-order power method for computing Z-eigenpairs of real symmetric tensors or a best rank-one approximation of real tensors. We illustrate our algorithms via numerical examples.

Bloch-Messiah reduction of Gaussian unitaries by Takagi factorization

The Bloch-Messiah (BM) reduction allows the decomposition of an arbitrarily complicated Gaussian unitary into a very simple scheme in which linear optical components are separated from nonlinear ones. The nonlinear part is due to the squeezing possibly present in the Gaussian unitary. The reduction is usually obtained by exploiting the singular value decomposition (SVD) of the matrices appearing in the Bogoliubov transformation of the given Gaussian unitary. This paper discusses a different approach, where the BM reduction is obtained in a straightforward way. It is based on the Takagi factorization of the (complex and symmetric) squeeze matrix and has the advantage of avoiding several matrix operations of the previous approach (polar decomposition, eigendecomposition, SVD, and Takagi factorization). The theory is illustrated with an application example in which the previous and present approaches are compared.

Complex-valued neural networks for the Takagi vector of complex symmetric matrices

This paper proposes complex-valued neural network for computing the Takagi vectors corresponding to the largest Takagi value of complex symmetric matrices. We establish some properties of the complex-valued neural network. Based on the Takagi factorization of complex symmetric matrices, we establish an explicit representation for the solution of the neural network and analyze its convergence property. Under certain conditions, we design a strategy to computing the Takagi factorization of a complex symmetric matrix by the proposed neural network. As an application, we consider the left and right singular vectors associated with the largest singular value for complex Toeplitz matrices. We illustrate our theory via numerical examples.

On the diagonalizability of a matrix by a symplectic equivalence, similarity or congruence transformation

A symplectic matrix S∈C2n×2n satisfies S=J−1STJ for J=[0−InIn0]∈R2n×2n. We will consider symplectic equivalence, similarity and congruence transformations and answer the question under which conditions a 2n×2n matrix is diagonalizable under one of these transformations. In particular, we will give symplectic analogues of the singular value decomposition and the Takagi factorization.

Singular value decomposition for the Takagi factorization of symmetric matrices

We describe a simple implementation of the Takagi factorization of symmetric matrices A=UΛUT with unitary U and diagonal Λ≥0 in terms of the square root of an auxiliary unitary matrix and the singular value decomposition of A. The method is based on an algebraically exact expression. For parameterized family Aε=A+εR=UεΛεUεT, ε≥0 with distinct singular values, the unitary matrices Uε are discontinuous at the point ε=0, if the singular values of A are multiple, but the composition UεΛεUεT remains numerically stable and converges to A. The factorization is represented as a fast and compact algorithm. Its demo version for Wolfram Mathematica and interactive numerical tests are available on Internet.

A comparison between the complex symmetric based and classical computation of the singular value decomposition of normal matrices

An algorithm for computing the singular value decomposition of normal matrices using intermediate complex symmetric matrices is proposed. This algorithm, as most eigenvalue and singular value algorithms, consists of two steps. It is based on combining the unitarily equivalence of normal matrices to complex symmetric tridiagonal form with the symmetric singular value decomposition of complex symmetric matrices. Numerical experiments are included comparing several algorithms, with respect to speed and accuracy, for computing the symmetric singular value decomposition (also known as the Takagi factorization). Next we compare the novel approach with the classical Golub-Kahan method for computing the singular value decomposition of normal matrices: it is faster, consumes less memory, but on the other hand the results are significantly less accurate.

A generalization of the complex Autonne–Takagi factorization to quaternion matrices

A complex symmetric matrix A can always be factored as A = UΣU T , in which U is complex unitary and Σ is a real diagonal matrix whose diagonal entries are the singular values of A. This factorization may be thought of as a special singular value decomposition for complex symmetric matrices. We present an analogous special singular value decomposition for a class of quaternion matrices that includes complex matrices that are symmetric or Hermitian.

On tridiagonal matrices unitarily equivalent to normal matrices

In this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied. It is well-known that any matrix is unitarily equivalent to a tridiagonal matrix. In case of a normal matrix the resulting tridiagonal inherits a strong relation between its super- and subdiagonal elements. The corresponding elements of the super- and subdiagonal will have the same absolute value. In this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied. Both an iterative reduction based on Krylov sequences as a direct tridiagonalization procedure via Householder transformations are reconsidered. This equivalence transformation is then applied to the normal case and equality of the absolute value between the super- and subdiagonals is proved. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved. Properties when applying the reduction on symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices and their relations with, e.g., complex symmetric and pseudo-symmetric matrices are presented. It is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the Takagi factorization. Finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given.

Hybrid Representations of Coupled Nonparametric and Parametric Models for Dynamic Systems

Parametric modeling of stochastic systems has proven useful for systems with well-defined and well-structured sources of uncertainty. The suitability of such models is usually indicated by small levels of uncertainty associated with their parameters. The parametric model may not be efficiently employed to deal with problems associated with a high level of uncertainty, particularly due to the modeling uncertainty. The class of so-called nonparametric stochastic models has recently been introduced to address this specific issue and found to be useful to some extent. This paper presents a coupling technique adapted to the receptance frequency-response-function matrix that will be useful for analyzing a complex dynamic system, particularly when it consists of several stochastic subsystems, each of which is individually deemed to be suitable for either a parametric model or a nonparametric model. Such a complex dynamic system is otherwise difficult to analyze. The existing nonparametric approach was, to date, applied to the real-valued positive-definite/semidefinite random system matrix: for example, mass, damping, and stiffness matrices. In the present work, the nonparametric approach is also employed with the complex-valued symmetric receptance frequency-response-function matrix, now acting as the system matrix, by having recourse to Takagi's factorization.

A twisted factorization method for symmetric SVD of a complex symmetric tridiagonal matrix

AbstractThis paper presents an O(n2) method based on the twisted factorization for computing the Takagi vectors of an n‐by‐n complex symmetric tridiagonal matrix with known singular values. Since the singular values can be obtained in O(n2) flops, the total cost of symmetric singular value decomposition or the Takagi factorization is O(n2) flops. An analysis shows the accuracy and orthogonality of Takagi vectors. Also, techniques for a practical implementation of our method are proposed. Our preliminary numerical experiments have verified our analysis and demonstrated that the twisted factorization method is much more efficient than the implicit QR method, divide‐and‐conquer method and Matlab singular value decomposition subroutine with comparable accuracy. Copyright © 2009 John Wiley & Sons, Ltd.

A Divide-and-Conquer Method for the Takagi Factorization

This paper presents a divide-and-conquer method for computing the symmetric singular value decomposition, or Takagi factorization, of a complex symmetric and tridiagonal matrix. An analysis of accuracy shows that our method produces accurate Takagi values and orthogonal Takagi vectors. Our preliminary numerical experiments have confirmed our analysis and demonstrated that our divide-and-conquer method is much more efficient than the implicit QR method even for moderately large matrices.