Generalisasi Invers Suatu Matriks Yang Memenuhi Persamaan Penrose

Generalized inverse is an extension of the concept of inverse matrix. One type of generalized inverse of a matrix of size (m × n) with elements of the complex number is the Moore Penrose inverse is denoted by A+ . Moore Penrose inverse is the inverse of the matrix which must satisfy the four equations called Penrose equations. Generalized Inverse whereas only satisfy some (not all) of the four Penrose equations are divided into classes based on the number of equations that can be met Penrose, {1}-inverse, {1,2}-inverse, {1,2.3}-inverse, dan {1,2,4}-inverse

Affine Generalized Inverse for Optimal Control Allocation

This research is a follow on to the "Optimal Control Prediction Method for Control Allocation" paper in which the Prediction Method iterative algorithm was introduced. Previously, the Prediction Method was shown to provide optimal control allocation solutions over the entire Attainable Moment Set for the Moore-Penrose and the generalized (weighted) inverse. As an extension to the Prediction Method, this paper introduces a family of Moore Penrose Affine Generalized Inverses, applicable for all moments, which compute control allocation solutions using a constant matrix and fixed null-space vector. The Moore-Penrose Affine Generalized Inverse is proven to yield equivalent solutions to those of the Prediction Method and therefore is guaranteed to yield Moore-Penrose optimal control allocation solutions. While the Prediction Method is applicable for any moment along an a priori specified moment direction, the Affine Generalized Inverse is shown to yield optimal control allocation solutions in a neighborhood of the given moment which is not restricted to a specified moment direction. Furthermore, the Affine Generalized Inverse is shown to provide the time derivative of optimal control allocation solutions and to facilitate maintaining solutions within control effector rate limitations. The Moore-Penrose Affine Generalized Inverse is broadened to encompass any arbitrary (weighted) Affine Generalized Inverse. Finally, a method of creating a moment lookup table is outlined to utilize the Affine Generalized Inverse as an offline control allocation solution for all moments in the Attainable Moment Set

Computation of generalized inverses by using the LDL∗ decomposition

AbstractAn efficient algorithm, based on the LDL∗ factorization, for computing {1,2,3} and {1,2,4} inverses and the Moore–Penrose inverse of a given rational matrix A, is developed. We consider matrix products A∗A and AA∗ and corresponding LDL∗ factorizations in order to compute the generalized inverse of A. By considering the matrix products (R∗A)†R∗ and T∗(AT∗)†, where R and T are arbitrary rational matrices with appropriate dimensions and ranks, we characterize classes A{1,2,3} and A{1,2,4}. Some evaluation times for our algorithm are compared with corresponding times for several known algorithms for computing the Moore–Penrose inverse

Application of Generalized Inverses and Circulant Matrices to Iterated Functions on Integers

After reviewing fundamental properties of matrix generalized inverses and circulant matrices, we describe a particular application in which both of these concepts play an important role. In particular, we establish the form of the Moore-Penrose generalized inverse of a type of circulant matrix that arises naturally in the study of iterated functions on integers

Generalizirani inverzi

Tema diplomskog rada je pojam generaliziranog inverza matrice. Prvo poglavlje se bavi Moore--Penroseovim inverzom i njegovom ulogom u rješavanju sustava linearnih jednadžbi. Nadalje, promotrili smo i $\{i, j, k\}$-inverze koji zadovoljavaju samo neke od Penroseovih uvjeta, svojstva koja zadovoljavaju i njihovu primjenu. Pokazane su i dvije metode za njegovo računanje, jedna koja se bazira na particioniranju matrice po stupcima i druga koja se bazira na dekompoziciji singularnih vrijednosti matrice. U drugom poglavlju smo promotrili Drazinov inverz i njegov specijalan slučaj, grupni inverz. Spomenute su primjene u teoriji Markovljevih lanaca i u sustavima diferencijalnih jednadžbi. Detaljnije su proučena tri algoritma za njegovo računanje i njihova konvergencija, te njihovo ponašanje na odabranim primjerima.This thesis’ emphasis has been on the theory and application of generalized inverses. The first part is about the Moore–Penrose inverse and its role in finding solutions of linear systems. $\{i, j, k\}$-inverse satisfies some, but not all, of the Penrose equations. Its properties and applications are studied. Two methods of computing the Moore–Penrose inverse are mentioned, one relying on partitioning the matrix by columns, and the other on the singular value decomposition. The second part was about the Drazin inverse and its properties. A special case of the Drazin inverse, the group inverse, was introduced and its role in the theory of finite Markov chains examined. An application to linear systems of differential equations was briefly discussed. Finally, three algorithms for its computing have been studied in some detail along with the convergence and some examples