The matrix sign function

Kenney, Charles; Laub, Alan J.

doi:10.1109/9.402226

Cited by 182 publications

(129 citation statements)

References 87 publications

Supporting

Mentioning

128

Contrasting

Unclassified

Order By: Relevance

“…We begin with a key connection between iterations for the matrix sign function and iterations for the canonical generalized polar decomposition. Recall that for a matrix A ∈ C n×n with no pure imaginary eigenvalues the sign function can be defined by sign(A) = A(A 2 ) −1/2 [8], [16]. The following theorem generalizes [12,Thm.…”

Section: Computational Considerationsmentioning

confidence: 99%

The Canonical Generalized Polar Decomposition

Higham¹,

Mehl²,

Tisseur³

2010

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

Abstract. The polar decomposition of a square matrix has been generalized by several authors to scalar products on R n or C n given by a bilinear or sesquilinear form. Previous work has focused mainly on the case of square matrices, sometimes with the assumption of a Hermitian scalar product. We introduce the canonical generalized polar decomposition A = W S, defined for general m × n matrices A, where W is a partial (M, N )-isometry and S is N -selfadjoint with nonzero eigenvalues lying in the open right half-plane, and the nonsingular matrices M and N define scalar products on C m and C n , respectively. We derive conditions under which a unique decomposition exists and show how to compute the decomposition by matrix iterations. Our treatment derives and exploits key properties of (M, N )-partial isometries and orthosymmetric pairs of scalar products, and also employs an appropriate generalized Moore-Penrose pseudoinverse. We relate commutativity of the factors in the canonical generalized polar decomposition to an appropriate definition of normality. We also consider a related generalized polar decomposition A = W S, defined only for square matrices A and in which W is an automorphism; we analyze its existence and the uniqueness of the selfadjoint factor when A is singular.

show abstract

Section: Computational Considerationsmentioning

confidence: 99%

The Canonical Generalized Polar Decomposition

Higham¹,

Mehl²,

Tisseur³

2010

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

show abstract

“…Roberts in [1] for the first time extended this definition for matrices, which has several important applications in scientific computing, for example see [2][3][4] and the references cited therein. For example, the off-diagonal decay of the matrix function of sign is also a well-developed area of study in statistics and statistical physics [5].…”

Section: Preliminariesmentioning

confidence: 99%

Constructing a High-Order Globally Convergent Iterative Method for Calculating the Matrix Sign Function

Jebreen

2018

Mathematical Problems in Engineering

View full text Add to dashboard Cite

This work is concerned with the construction of a new matrix iteration in the form of an iterative method which is globally convergent for finding the sign of a square matrix having no eigenvalues on the axis of imaginary. Toward this goal, a new method is built via an application of a new four-step nonlinear equation solver on a particulate matrix equation. It is discussed that the proposed scheme has global convergence with eighth order of convergence. To illustrate the effectiveness of the theoretical results, several computational experiments are worked out. PreliminariesThe sign function for the scalar case is expressed bywherein ∈ C is not located on the imaginary axis. Roberts in [1] for the first time extended this definition for matrices, which has several important applications in scientific computing, for example see [2][3][4] and the references cited therein. For example, the off-diagonal decay of the matrix function of sign is also a well-developed area of study in statistics and statistical physics [5].To proceed formally, let us consider that ∈ C × is a square matrix possessing no eigenvalues on the axis of imaginary. We consideras a form of Jordan canonical written such that wherein + = . Noting that sign( ) is definable once is nonsingular. This procedure takes into account a clear application of the form of Jordan canonical and of the matrix . Herein none of the matrices and are unique. However, it is possible to investigate that sign( ) as provided in (4) does not rely on the special selection of or .Here, a simpler interpretation for the sign matrix in the case of Hermitian case (namely, all eigenvalues are real) can be given byis a diagonalization of the square matrix . The significance of calculating and finding in (4) is because of the point that the function of sign plays a central role in matrix functions theory, specially for principal matrix

show abstract

“…This iteration arises from replacing Z −1 k in (4) by the Schulz iteration for the inverse of a matrix [43] and has been studied for sign function computations, e.g., in [5,39,40]. It can be interpreted as the matrix version of recursively applying the "rational" (here: polynomial) function…”

Section: The Newton-schulz Iterationmentioning

confidence: 99%

“…The matrix sign function of Z is defined as sign (Z) := S −I k 0 0 I n−k S −1 . Many other definitions of the sign function can be given; see [40] for an overview. Some important properties of the matrix sign function are summarized in the next lemma.…”

Section: Theoretical Backgroundmentioning

confidence: 99%