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Square matrix
About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.
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TL;DR: In this article, an efficient numerical solution of the matrix equation $AX^2 + BX + C = 0, where A, B, C and X are all square matrices, is presented.
Abstract: This paper is concerned with the efficient numerical solution of the matrix equation $AX^2 + BX + C = 0$, where A, B, C and X are all square matrices. Such a matrix X is called a solvent. This equation is very closely related to the problem of finding scalars $\lambda $ and nonzero vectors x such that $(\lambda ^2 A + \lambda B + C)x = 0$. The latter equation represents a quadratic eigenvalue problem, with each $\lambda $ and x called an eigenvalue and eigenvector, respectively. Such equations have many important physical applications.By presenting an algorithm to calculate solvents, we shall show how the eigenvalue problem can be solved as a byproduct. Some comparisons are made between our algorithm and other methods currently available for solving both the solvent and eigenvalue problems. We also study the effects of rounding errors on the presented algorithm, and give some numerical examples.
73 citations
TL;DR: In this article, a slightly relaxed version of Yang's result is presented, using a different method from the one we use in this paper, which is based on the one used in the present paper.
73 citations
TL;DR: Following simple geometric arguments, a natural and geometrically meaningful definition of square matrices is derived and this definition is applied to matrices in computer graphics.
Abstract: Geometric transformations are most commonly represented as square matrices in computer graphics. Following simple geometric arguments we derive a natural and geometrically meaningful definition of ...
73 citations
TL;DR: In this article, it was shown that the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues and that the set can be obtained as the intersection of closed half planes (of complex numbers).
Abstract: Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of closed half planes (of complex numbers). As a result, it is always a convex set in $\mathcal C$. Moreover, the higher rank numerical range of a normal matrix is a convex polygon determined by the eigenvalues. These two consequences confirm the conjectures of Choi et al. on the subject. In addition, the results are used to derive a formula for the optimal upper bound for the dimension of a totally isotropic subspace of a square matrix, and verify the solvability of certain matrix equations.
73 citations
TL;DR: In this article, conditions for the existence of solutions of linear matrix equations are given, when it is known a priori that the solution matrix has a given structure (e.g. symmetric, triangular, diagonal).
Abstract: Conditions for the existence of solutions, and the general solution of linear matrix equations are given, when it is known a priori that the solution matrix has a given structure (e.g. symmetric, triangular, diagonal). This theory is subsequently extended to matrix equations that are linear in several unknown ‘structured’ matrices, and to partitioned matrix equations.
72 citations