A Note on Sin Θ Theorems for Singular Subspace Variations

New perturbation theorems for bases of singular subspaces are proved. These theorems complement the known sin theorems for singular subspace perturbations, taking into account a kind of sensitivity of singular vectors discarded by previous theorems. Furthermore these results guarantee that high relative accuracy algorithms for the SVD are able to compute reliably simultaneous bases of left and right singular subspaces.     

Perturbation analysis of generalized singular subspaces

This paper, as a continuation of the paper [20] in Numerische Mathematik, studies the subspaces associated with the generalized singular value decomposition. Second order perturbation expansions, Fréchet derivatives and condition numbers, and perturbation bounds for the subspaces are derived. Received January 26, 1996 / Revised version received May 14, 1997     

Bounds for Eigenvalues of Matrix Polynomials Over Quaternion Division Algebra

Localization theorems are discussed for the left and right eigenvalues of block quaternionic matrices. Basic definitions of the left and right eigenvalues of quaternionic matrices are extended to quaternionic matrix polynomials. Furthermore, bounds on the absolute values of the left and right eigenvalues of quaternionic matrix polynomials are devised and illustrated for the matrix p norm, where \({p = 1, 2, \infty, F}\). The above generalizes the bounds on the absolute values of the eigenvalues of complex matrix polynomials, which give sharper bounds to the bounds developed in [LAA, 358, pp. 5–22 2003] for the case of 1, 2, and \({\infty}\) matrix norms.     

Perturbation analysis of singular subspaces and deflating subspaces

Summary. Perturbation expansions for singular subspaces of a matrix and for deflating subspaces of a regular matrix pair are derived by using a technique previously described by the author. The perturbation expansions are then used to derive Fr\'echet derivatives, condition numbers, and th-order perturbation bounds for the subspaces. Vaccaro's result on second-order perturbation expansions for a special class of singular subspaces can be obtained from a general result of this paper. Besides, new perturbation bounds for singular subspaces and deflating subspaces are derived by applying a general theorem on solution of a system of nonlinear equations. The results of this paper reveal an important fact: Each singular subspace and each deflating subspace have individual perturbation bounds and individual condition numbers. Received July 26, 1994     

The embedding theorem in limiting cases and its applications in singular perturbation

In 1980, Brézis^[6], using the technique of dividing the total space into two parts, proved the embedding theorem of limiting case which is very important in applications. In 1982, Ding Xiaqi improved the proof given in [6], by using of the technique of dividing the total space into three parts. In this paper, using the technique of dividing the total space into three parts, the author proves uniformly the results obtained by Ding^[3,4], and gives an embedding theorem of limiting case including (Lemma 2.2). And he also gives two kinds of examples, applying the embedding theorems (limiting case and non-limiting case) and the interpolation theorems. These examples are the singular perturbation problems in the sense of Lions^[1] (for the definition of singular perturbation, see [1], Introduction). But the singular solutionU _e converges uniformly to the limit solution (degenerate)U, ase0.     

Singularity, Wielandt’s lemma and singular values

In this study, some upper and lower bounds for singular values of a general complex matrix are investigated, according to singularity and Wielandt’s lemma of matrices. Especially, some relationships between the singular values of the matrix A and its block norm matrix are established. Based on these relationships, one may obtain the effective estimates for the singular values of large matrices by using the lower dimension norm matrices. In addition, a small error in Piazza (2002) [G. Piazza, T. Politi, An upper bound for the condition number of a matrix in spectral norm, J. Comput. Appl. Math. 143 (1) (2002) 141-144] is also corrected. Some numerical experiments on saddle point problems show that these results are simple and sharp under suitable conditions.     

矩阵广义逆新的扰动上界

利用矩阵的奇异值分解方法,研究了矩阵广义逆的扰动上界,得到了在F-范数下矩阵广义逆的扰动上界定理,所得定理推广并彻底改进了近期的相关结果.相应的数值算例验证了定理的有效性.     

Some remarks on the perturbation of polar decompositions for rectangular matrices

In this article we focus on perturbation bounds of unitary polar factors in polar decompositions for rectangular matrices. First we present two absolute perturbation bounds in unitarily invariant norms and in spectral norm, respectively, for any rectangular complex matrices, which improve recent results of Li and Sun (SIAM J. Matrix Anal. Appl. 2003; 25 :362–372). Secondly, a new absolute bound for complex matrices of full rank is given. When ‖A ? Ã‖₂ ? ‖A ? Ã‖_F, our bound for complex matrices is the same as in real case. Finally, some asymptotic bounds given by Mathias (SIAM J. Matrix Anal. Appl. 1993; 14 :588–593) for both real and complex square matrices are generalized. Copyright © 2005 John Wiley & Sons, Ltd.     

Singular stationary measures are not always fractal

We know of few explicit results to insure that stationary measures are simultaneously (i) singular, (ii) nonatomic, (iii) with interval support, and (iv) unique. Such results would appear useful, to further separate the analytic notion of singular from the geometric notion of fractal. We prove two general theorems, one for maps of [0,1] into [0, 1], the other for 2×2 random matrices. In each setting, we study measures supported on two points of the transformation space, and we provide sufficient conditions to insure that the stationary measures satisfy (i)–(iv).     

Relative eigenvalue and singular value perturbations of scaled diagonally dominant matrices

The paper derives improved relative perturbation bounds for the eigenvalues of scaled diagonally dominant Hermitian matrices and new relative perturbation bounds for the singular values of symmetrically scaled diagonally dominant square matrices. The perturbation result for the singular values enlarges the class of well-behaved matrices for accurate computation of the singular values. AMS subject classification (2000)  65F15     

Certain exact bounds for seminorms given on spaces of periodic functions

Certain new bounds are established for the values of seminorms given on the spaces C and L^p (1p<) of periodic functions by means of the norm of the function itself and its finite differences, as well as of the moduli of continuity. These bounds are applied to concrete seminorms; in particular, to the best approximation, which yields a refinement of the direct theorems in approximation theory. The results obtained for spaces C and L¹ are exact.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 789–798, June, 1977.     

Backward perturbation analysis of certain characteristic subspaces

Summary This paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemAx=x, the generalized eigenvalue problem Ax=Bx, and the singular value decomposition of a matrixA. This paper also gives residual bounds for certain eigenvalues, generalized eigenvalues and singular values.This subject was supported by the Swedish Natural Science Research Council and the Institute of Information Processing of the University of Umeå.     

An optimal perturbation bound

Cai and Zhang establish separate perturbation bounds for distances with spectral and Frobenius norms (Cai T, Zhang A. Rate‐optimal perturbation bounds for singular subspaces with applications to high‐dimensional statistics. The Annals of Statistics. 2018; Vol. 46, No. 1: 60?89). We extend their theorem to each unitarily invariant norm. It turns out that our estimation is optimal as well.     

Low Rank Approximation of a Hankel Matrix by Structured Total Least Norm

The structure preserving rank reduction problem arises in many important applications. The singular value decomposition (SVD), while giving the closest low rank approximation to a given matrix in matrix L ₂ norm and Frobenius norm, may not be appropriate for these applications since it does not preserve the given structure. We present a new method for structure preserving low rank approximation of a matrix, which is based on Structured Total Least Norm (STLN). The STLN is an efficient method for obtaining an approximate solution to an overdetermined linear system AX B, preserving the given linear structure in the perturbation [E F] such that (A + E)X = B + F. The approximate solution can be obtained to minimize the perturbation [E F] in the L _p norm, where p = 1, 2, or . An algorithm is described for Hankel structure preserving low rank approximation using STLN with L _p norm. Computational results are presented, which show performances of the STLN based method for L ₁ and L ₂ norms for reduced rank approximation for Hankel matrices.     

On the number of solutions of norm form equations

In 1985 Evertse and Gyory [5] gave explicit upper bounds for the number of solutions of norm form equations of the form (1.1) under the hypotheses that (i) x m &nequiv; 0, &alpha; 1 = 1, &alpha; 2 , ... ,&alpha; m-1 are Q-linearly independent and has degree at least 3 overQ( &alpha; 1 ..., &alpha; m-1 ), or that (ii) the degree of i is at least 3 over Q(&alpha; 1 , ..., &alpha; i-1 ) for i = 2, m. Later Gyry [9], Evertse [3] and Evertse and Gy}ory [6] derived general upper bounds for arbitrary norm form equations which include the case (ii), but not the case (i). In the present paper we considerably improve the bounds of [5], and we give a further improvement which is valid for all but at most finitely many possible values of the constant term b of the equation. Our bound obtained under the assumption (ii) is better for almost all b than the general bounds of [9], [3] and [6].     

The point spectra for certain classes of operators in B(ℓ p )

In a recent paper [5] Maddox determined the point spectrum of the Cesaro matrices of order >0, considered as series-to-series operators. In this paper we obtain the point spectrum for the Endl generalized Hausdorff matrices, the Hausdorff matrices, and for a wide class of generalized Hausdorff matrices, which include the Endl and ordinary Hausdorff matrices as special cases. We also obtain point spectrum results for a wide class of weighted mean matrices. Since the Cesaro matrices are examples of Hausdorff matrices, our results contain the corresponding theorems of Maddox as special cases.     

Eigenvalues of symmetrizable matrices

New perturbation theorems for matrices similar to Hermitian matrices are proved for a class of unitarily invariant norms calledQ-norms. These theorems improve known results in certain circumstances and extend Lu's theorems for the spectral norm, see [Numerical Mathematics: a Journal of Chinese Universities, 16 (1994), pp. 177–185] toQ-norms. This material is based in part upon the third author's work supported from August 1995 to December 1997 by a Householder Fellowship in Scientific Computing at Oak Ridge National Laboratory, supported by the Applied Mathematical Sciences Research Program, Office of Energy Research, United States Department of Energy contract DE-AC05-96OR22464 with Lockheed Martin Energy Research Corp.     

On the condition number of the total least squares problem

This paper concerns singular value decomposition (SVD)-based computable formulas and bounds for the condition number of the total least squares (TLS) problem. For the TLS problem with the coefficient matrix $A$ and the right-hand side $b$ , a new closed formula is presented for the condition number. Unlike an important result in the literature that uses the SVDs of both $A$ and $[A,\ b]$ , our formula only requires the SVD of $[A,\ b]$ . Based on the closed formula, both lower and upper bounds for the condition number are derived. It is proved that they are always sharp and estimate the condition number accurately. A few lower and upper bounds are further established that involve at most the smallest two singular values of $A$ and of $[A,\ b]$ . Tightness of these bounds is discussed, and numerical experiments are presented to confirm our theory and to demonstrate the improvement of our upper bounds over the two upper bounds due to Golub and Van Loan as well as Baboulin and Gratton. Such lower and upper bounds are particularly useful for large scale TLS problems since they require the computation of only a few singular values of $A$ and $[A, \ b]$ other than all the singular values of them.     

不变子空间与广义不变子空间——(Ⅱ)扰动定理

关于矩阵的不变子空间,自然会提出这样一个扰动问题:设Z_1∈C~(n×l)是A∈C~(n×n)的一个特征矩阵,若E∈C~(n×n)是一个扰动矩阵,问A+B是否存在特征矩阵Z_1,使得(Z_1)靠近R(Z_1)?关于矩阵对的广义不变子空间.也可以类似地提出问题。 对于这些问题,G.W.Stewart曾经讨论过,他的方法的关键是构造一种求解二次矩阵方程的迭代过程,用来逼近矩阵的一个不变子空间;而本文建议另一种迭代格式,用这种迭代逼近一个不变(或广义不变)子空间,具有二次收敛速度。