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A multiplicative perturbation M of a matrix T has the form M=ETF∗, where E and F are square matrices. It is proved that every acute perturbation is essentially a strong perturbation, which is a typ...

For a matrix A, let A † denote its Moore–Penrose inverse. A matrix M is called a multiplicative perturbation of T ∈ C m × n if M = E T F ⁎ for some E ∈ C m × m and F ∈ C n × n . Based on the alternative expression for M as M = ( E T T † ) ⋅ T ⋅ ( F T † T ) ⁎ , the generalized triple reverse order law for the Moore–Penrose inverse is obtained as M † = ( ( F T † T ) ⁎ ) † ⋅ ( Y Y † T Z Z † ) L R − 1 † ⋅ ( E T T † ) † , where ( Y Y † T Z Z † ) L R − 1 † is the weighted Moore–Penrose inverse for certain matrices Y , Z , L and R associated to the triple ( T , E , F ) . Furthermore, it is proved that this weighted Moore–Penrose inverse in the resulting expression for M † can be really replaced with T † if ( E T T † ) † E T T † ⋅ T = T ⋅ ( F T † T ) † ( F T † T ) . In the special case that rank ( M ) = rank ( T ) or M is a weak perturbation of T, a simplified version of M † , as well as M M † and M † M , is also derived.

It is well known that the Cramer’s rule for the solution \(\mathbf {x}\) of a nonsingular equation

Before Moore introduced the generalized inverse of matrices by algebraic methods, Fredholm, Hilbert, Schmidt, Bounitzky, Hurwitz and other mathematicians had studied the generalized inverses of integral operators and differential operators. Recently, due to the development of science and technology and the need for practical problems, researchers are very interested in the study of the generalized inverses of linear operators in abstract spaces.

Let A be a given matrix. When computing a generalized inverse of A, due to rounding error, we actually obtain the generalized inverse of a perturbed matrix \(B=A+E\) of A. It is natural to ask if the generalized inverse of B is close to that of A when the perturbation E is sufficiently small.

A polynomial matrix is a matrix whose entries are polynomials. Equivalently, a polynomial matrix can be expressed as a polynomial with matrix coefficients. Formally speaking, in the univariable case, \((\mathbb {R}[x])^{m \times n}\) and \((\mathbb {R}^{m \times n}) [x]\) are isomorphic. In other words, extending the entries of matrices to polynomials is the same as extending the coefficients of polynomials to matrices.

The reverse order law for the generalized inverses of a matrix product yields a class of interesting fundamental problems in the theory of the generalized inverses of matrices. They have attracted considerable attention since the middle 1960s.

Let X be a Hilbert space and L(X) be the vector space of the linear operators from X into X. We denote the set of bounded linear operators from X into X by B(X). In this chapter, we will investigate the definition, basic properties, representation theorem and computational methods for the Drazin inverse of an operator \(T \in B(X)\), \(\mathcal {R}(T^k)\) is closed, where \(k=\mathrm {Ind}(T)\) is the index of T.

There are various ways to introduce the generalized inverses. We introduce them by considering the problem of solving systems of linear equations.

Having studied the perturbation of the M-P inverse and the weighted M-P inverse, we now turn to the perturbation analysis of the Drazin and group inverses.

A matrix is considered structured if its structure can be exploited to obtain efficient algorithms. Examples of structured matrices include Toeplitz, Hankel, circulant, Vandermonde, Cauchy, sparse. A matrix is called Toeplitz if its entries on the same diagonal are equal.

In this paper, we first give the finite algorithm for generalized inverse A"T","S^(^2^) of a matrix A over an integral domain, and, based on it and the discrete Fourier transform, present an algorithm for calculating {2}-inverses of a polynomial matrix with prescribed image and kernel. And the algorithm is implemented in the Mathematica programming language and expands the algorithms in [13].

In this paper we establish the definition of the generalized inverse A(2)T, Swhich is a {2} inverse of a matrixAwith prescribed imageTand kernelsover an associative ring, and give necessary and sufficient conditions for the existence of the generalized inverseand some explicit expressions forof a matrix A over an associative ring, which reduce to the group inverse or {1} inverses. In addition, we show that for an arbitrary matrixAover an associative ring, the Drazin inverse Ad, the group inverse Agand the Moore-Penrose inverse. if they exist, are all the generalized inverse A(2)T, S.

In this paper, we study the idempotency of the generalized Schur complement on generalized inverse A T , S ( 2 ) , together with other familiar generalized inverses such as A M , N † , A † , A d and A g , defined from a partitioned idempotent matrix, which is a complementarity of [Jerzy K. Baksalary, Oskar Maria Baksalary, Tomasz Szulc, Properties of Schur complements in partitioned idempotent matrices, Lin. Alg. Appl. 379 (2004) 303–318; D.V. Ouellette, Schur complememnts and statistics, Lin. Alg. Appl. 36 (1981) 187–295].

LetA andE bem x n matrices andW an n xm matrix, and letAd,W denote the W-weighted Drazin inverse ofA. In this paper, a new representation of the W-weighted Drazin inverse ofA is given. Some new properties for the W-weighted Drazin inverseAd,W and Bd,W are investigated, whereB =A+E. In addition, the Banach-type perturbation theorem for the W-weighted Drazin inverse ofA andB are established, and the perturbation bounds for ∥Bd,W∥ and ∥Bd, W, -Ad,W∥/∥Ad,W∥ are also presented. WhenA andB are square matrices andW is identity matrix, some known results in the literature related to the Drazin inverse and the group inverse are directly reduced by the results in this paper as special cases.

A Cramer rule for solving a kind of restricted matrix equationsWAWXW~BW~=D,R(X)@?R[(AW)^k^"^1],N(X)@?N[(W~B)^k^"^2^@?]is presented in [G. Wang, Jie Sun, A Cramer rule for solution of the general restricted matrix equation, Appl. Math. Comput. 154 (2004) 415-422]. This paper gives a more condensed Cramer rule for the solution of the matrix equations. The results in [G. Wang, Z.L. Xu, Solving a kind of restricted matrix equations and Cramer rule, Appl. Math. Comput. 162 (2005) 329-338, Chao Gu, G.R. Wang, Z.L. Xu, PCR Algorithm for the Parallel Computation of the Solution of a class of Singular linear Systems, Appl. Math. Comput. 176 (2006) 237-244] are partially the special cases in our paper.

In this paper, the transformations of generating systems of Euclidean space are examined in connection with the 2-inverseA T,S (2) , S which has prescribed rangeT and null spaceS of their Gram matrices. The biorthogonal systems of the Moore-Penrose inverse and the Drazin inverse are extended to the {2}-inverseA T,S (2) .

By using the rank methods of matrix, a necessary and sufficient condition is established for reverse order law $$\begin{gathered} WA_{d,W} W = (W_n (A_n )_{d,W_n } W_n )(W_{n - 1} (A_{n - 1} )_{d,W_{n - 1} } W_{n - 1} ) \hfill \\ ... (W_1 (A_1 )_{d,W_1 } W_1 ) \hfill \\ \end{gathered} $$ to hold for the W-weighted Drazin inverses, whereA =A1A2… An andW =WnWn-1 …W1. This result is the extension of the result proposed by [Linear Algebra Appl., 348(2002)265-272] and the result proposed by [J. Math. Research and Exposition. 19(1999)355-358].

In this article, we establish the definition of the generalized inverse , which is {2} inverse of matrix A with prescribed image T and kernel S over a commutative ring R, and give an explicit expression for over integral domains, which generalizes the explicit expression for over the field of complex numbers [Wei Yimin, 1998, A characterization and representation of the generalized inverse and its applications, Linear Algebra Applications, 280, 87–96, Theorem 2.1]. In addition, we show that over integral domains, the Drazin inverse, the group inverse and the Moore–Penrose inverse are all .

In this paper, we introduce a PQ-norm and discuss the condition number of the W-weighted Drazin inverse. We get some results on the condition number with respect to A"d","w and singular linear systems WAWx=b,b@?R((WA)^k^"^2), x@?R((AW)^k^"^1) which are similar to the regular inverse A^-^1 and nonsingular linear systems Ax=b.

The problem of finding determinantal formula for solutions of some restricted linear systems Ax=b has been discussed in [Linear Multilinear Algebra 15 (1984) 319, Linear Algebra Appl. 116 (1989) 27, Linear Multilinear Algebra 34 (1993) 177, Appl. Math. Comput. 125 (2002) 303]. This paper deals with some more extensive cases of this kind of problem and establishes determinantal formulas for solutions of the restricted matrix equationsAX=D(R(X)@?R(A^k^"^1)),XB=D(N(X)@?N(B^k^"^2)),AXB=D(R(X)@?R(A^k^"^1),N(X)@?N(B^k^"^2)),where A@?C^n^x^n with Ind(A)=k"1, B@?C^m^x^m with Ind(B)=k"2, and D@?C^n^x^m. The results in [Linear Multilinear Algebra 15 (1984) 319, Linear Algebra Appl. 74 (1986) 213, Linear Multilinear Algebra 34 (1993) 177, Appl. Math. Comput. 125 (2002) 303] are partially the special cases in our paper. The classic Cramer rule is also a special case of our results.

A Leverrier-like algorithm is presented which allows the computation of the transfer function of a linear regular system from its 2-D state-space description, without inverting a multivariable polynomial matrix. This algorithm is an extension of the classic Leverrier's algorithm for 1-D system and it reduces the computational cost. 2-D Cayley-Hamilton theorem is also shown by the algorithm.

This paper presents the solution of a general constrained matrix equation using generalized inverses and gives an explicit expression for the elements of the solution matrix using Cramer rule.

By using the ranks of matrices, the necessary and sufficient condition is given for the reverse order law of the generalized inverse A^(^2^)"T","S of matrix product A=A"1A"2...A"n to hold. As the special cases, similar results for Moore-Penrose inverse A^@?, the weighted Moore-Penrose inverse A^@?"M","N, Drazin inverse A"d and group inverse A"g are also derived.

A constructive proof for the recursive representation of the weighted Moore-Penrose inverse of a partitioned matrix A=(UV) is given. Using the same thread of reasoning, similar results for A^(^1^,^3^M^) and A^(^1^,^4^N^) are derived. And all of these results can be represented by an unified form.

A Cramer rule for finding the unique W-weighted Drazin inverse solution of a class of restricted matrix equationsWAWXWBW=D,R(X)@?R[(AW)^k^"^1],N(X)@?N[(WB)^k^@?^"^2]is presented, and the results in [Linear Algebra Appl. 116 (1989) 27, Appl. Math. Comput. 125 (2002) 303] can be deduced from this paper. Where [email protected]?C^m^x^n, [email protected]?C^n^x^m, Ind(AW)=k"1, [email protected]?C^p^x^q, [email protected]?C^q^x^p, Ind(WB)[email protected]?"2, and [email protected]?C^n^x^p.

This paper studies the continuity of the generalized inverse A"T","S^(^2^). Since common important generalized inverses, such as A^+, A"M","N^+, A"d, A^#, A"d","w, A"("L")^(^-^1^) and A"("L")^(^+^) etc. are all generalized inverse A"T","S^(^2^). In fact, we give a unified continuity of such important generalized inverses.

This paper presents a new highly parallel algorithm for computing the minimum-norm (T) least-squares (S) solution of inconsistent linear equations Ax=b([email protected]?R^m^x^n"r,[email protected]?R(A)). By this algorithm the solution x=A"S","T^+b is obtained in T=(1+m)(1+log"2m)+n(6+log"2(n-r+1)+log"2m+log"2n)-r(1+log"2n) steps with P=mn processors when m>=2(n-1) and with P=2n(n-1) processors otherwise.

By using the ranks of matrices, the necessary and sufficient condition is given for the reverse order law ( A 1 A 2 ⋯ A n ) D = A n D A n −1 D ⋯ A 2 D A 1 D to hold.

The Leverrier–Chebyshev algorithm is presented. It is an extension of the Leverrier–Fadeev algorithm for simultaneous computation of the adjoint B(μ) and determinant a(μ) of the singular pencil μE−A, where E is singular, but det(μE−A)≢0. We express the B(μ) and a(μ) relative to a basis of orthogonal Chebyshev polynomial, and as such solve a problem of S. Barnett.

Some additive perturbation results for Drazin inverses are given. In particular, a formula is given for the Drazin inverse of a sum of two matrices, when one of the products of these matrices vanishes. Some special applications of this are also considered.

An algorithm for computing the inverse of a matrix polynomial s N J - s N -1A 1 -⋯- sA N-1 - A N is given, where J is a singular matrix. This algorithm is a new extension of Leverrier's algorithm for computing the inverse of a matrix polynomial s N I - s N -1 A 1 -⋯- sA N -1- A N , where I denotes the unit matrix.

A new modelling method of image Jacobian estimation is presented for uncalibrated visual servoing of robots, in which a support vector regression(SVR) technique is used for non-linear mapping between target image features and robot joint angles, and an image Jacobian expression is derived from the SVR equation with Gaussian kernel. The experiments of robot visual servoing with both eye-in-hand and eye-to-hand camera configuration are conducted using the SVR-Jacobian estimator which has been automatically trained by self-learning under the control of computer, two kinds of experimental results have shown that the robot visual servoing converges at the desired goal and high target-tracking accuracy can be acquired with the proposed modelling method.

We give an improved parallel algorithm for computing the generalized inverse A+ by using some results of F.P. Preparata and D.V. Sarwate (1978) and modifying the algorithm of G. Wang and S. Lu (1988). We show that, with the same assumption as Preparata and Sarwate, the time complexity and the number of processors using the improved parallel algorithm are T=O(logr log n) and P=max {⌈m/n⌉nα/logn, 2r1/2nα/(log r log n) }, respectively. The cost-optimality, P ∗ T is less than P ∗ T (T and P are the time complexity and the number of processors using the orginal parallel algorithm).

A general epipolar line constraint equation for binocolar stereo vision is first established for the measurement of workpiece surface, in which the intrinsic parameters of two cameras and their extrinsic structural parameters with respect to a reference base are included. To overcome the drawback of the relative distortion between left image and right image, An epipolar line rectification technique is used for the corresponding point detection in the stereovision-based measurement system. Instead of traditional rectangular window, a circular window sliding on epipolar lines is used to compute the correlation coefficient between the regions surrounding a selected point in the left image and its corresponding point in the right image. The stereovision-based measurement of a T-shaped circular pipe (as an example of workpieces) is carried out using the rectified epipolar lines and the workpiece surface is reconstructed with its acquired three-dimensional coordinates. Experimental results show that the relative distortion of left and right images can be eliminated and the measurement accuracy can be greatly improved after the application of epipolar line rectification, and the workpiece surface is reconstructed with less errors.

Fuzzy enhancement algorithm and KSW entropy algorithm are proposed for pre-welding seam image segmentation. Before automatic welding, the welding seam image is often segmented so that it can be recognized by robot. KSW entropy algorithm is an effective way for segmentation. Image enhancement is often applied before segmentation in order to get better result. As the pre-welding seam image is not uniform, the effect is not ideal after applying regular enhancement algorithm and KSW entropy algorithms to it. Fuzzy enhancement algorithm can eliminate the noises and preserve the details of the image intelligently. The image dividing point has close relation with the result of fuzzy enhancement algorithm. Self adapting algorithm about how to get the proper image dividing point is discussed. When fuzzy enhancement algorithm and KSW entropy segmentation algorithm are used to pre-welding seam image, the experiment results prove that the best effect can be got and little time will be cost.

The representations for the weighted Moore-Penrose inverse, the Drazin inverse, and the group inverse of the Kronecker product A ⊗ B of the two matrices A and B are given, and by using these results, the determinant forms for projectors AA + MN , A + MN A , A d A , and A g A are deduced.

In this paper, the minimum-norm (T) least-squares (S) solution of inconsistent linear equations Ax = y(AϵC m×n r ,y∉R(A)) is given a determinantal form and reduces to common Cramer's rule if A is nonsingular.