Topic

# Square matrix

About: Square matrix is a(n) research topic. Over the lifetime, 5000 publication(s) have been published within this topic receiving 92428 citation(s).

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TL;DR: In this article, a matrix of rank two can be represented as a biplot, which consists of a vector for each row and a column, chosen so that any element of the matrix is exactly the inner product of the vectors corresponding to its row and to its column.

Abstract: SUMMARY Any matrix of rank two can be displayed as a biplot which consists of a vector for each row and a vector for each column, chosen so that any element of the matrix is exactly the inner product of the vectors corresponding to its row and to its column. If a matrix is of higher rank, one may display it approximately by a biplot of a matrix of rank two which approximates the original matrix. The biplot provides a useful tool of data analysis and allows the visual appraisal of the structure of large data matrices. It is especially revealing in principal component analysis, where the biplot can show inter-unit distances and indicate clustering of units as well as display variances and correlations of the variables. Any matrix may be represented by a vector for each row and another vector for each column, so chosen that the elements of the matrix are the inner products of the vectors representing the corresponding rows and columns. This is conceptually helpful in understanding properties of matrices. When the matrix is of rank 2 or 3, or can be closely approximated by a matrix of such rank, the vectors may be plotted or modelled and the matrix representation inspected physically. This is of obvious practical interest for the analysis of large matrices. Any n x m matrix Y of rank r can be factorized as

2,498 citations

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Rice University

^{1}TL;DR: The iterative scheme is shown to be a truncation of the standard implicitly shifted QR-iteration for dense problems and it avoids the need to explicitly restart the Arnoldi sequence.

Abstract: The Arnoldi process is a well-known technique for approximating a few eigenvalues and corresponding eigenvectors of a general square matrix. Numerical difficulties such as loss of orthogonality and assessment of the numerical quality of the approximations, as well as a potential for unbounded growth in storage, have limited the applicability of the method. These issues are addressed by fixing the number of steps in the Arnoldi process at a prescribed value k and then treating the residual vector as a function of the initial Arnoldi vector. This starting vector is then updated through an iterative scheme that is designed to force convergence of the residual to zero. The iterative scheme is shown to be a truncation of the standard implicitly shifted QR-iteration for dense problems and it avoids the need to explicitly restart the Arnoldi sequence. The main emphasis of this paper is on the derivation and analysis of this scheme. However, there are obvious ways to exploit parallelism through the matrix-vector ...

1,112 citations

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TL;DR: A special class of indefinite quadratic programs is constructed, with simple constraints and integer data, and it is shown that checking (a) or (b) on this class is NP-complete.

Abstract: In continuous variable, smooth, nonconvex nonlinear programming, we analyze the complexity of checking whether(a)a given feasible solution is not a local minimum, and(b)the objective function is not bounded below on the set of feasible solutions.
We construct a special class of indefinite quadratic programs, with simple constraints and integer data, and show that checking (a) or (b) on this class is NP-complete. As a corollary, we show that checking whether a given integer square matrix is not copositive, is NP-complete.

1,002 citations

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TL;DR: A new method, called the QZ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B with particular attention to the degeneracies which result when B is singular.

Abstract: A new method, called the $QZ$ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B. Particular attention is paid to the degeneracies which result when B is singular. No inversions of B or its submatrices are used. The algorithm is a generalization of the $QR$ algorithm, and reduces to it when $B = I$. Problems involving higher powers of $\lambda $ are also mentioned.

992 citations