Square matrix

About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.

Physical interpretation of signal reconstruction from reduced rank matrices

Abstract: An enhanced version of a signal is obtained after subtraction of a linear combination of rank one matrices from the signal observation matrix. This operation results in a nonToeplitz observation matrix. By averaging the elements of this matrix, a new Toeplitz matrix is produced and a reconstructed signal is generated. This averaging operation is examined, and physical interpretations are given. >

Square roots of operators

Abstract: Introduction. If H is a complex Hilbert space and if A is an operator on H (i.e., a bounded linear transformation of H into itself), under what conditions does there exist an operator B on H such that B2=A? In other words, when does an operator have a square root? The spectral theorem implies that the normality of A is a sufficient condition for the existence of B; the special case of positive definite operators can be treated by more elementary means and is, in fact, often used as a step in the proof of the spectral theorem. As far as we are aware, no useful necessary and sufficient conditions for the existence of a square root are known, even in the classical case of finite-dimensional Hilbert spaces. The problem of finding some easily applicable conditions is of interest, in part because the use of square roots is frequently a helpful technique in the study of algebraic properties of operators, and in part because of the information that such conditions might yield about the hitherto rather mysterious behavior of non-normal operators. If a non-zero, 2-rowed square matrix is nilpotent, then its index of nilpotence is equal to 2; this comment shows that no such matrix can have a square root. On the other hand, an elementary computation, based on the Jordan canonical form, shows that every invertible matrix does have a square root. Since the number 0 is known to have a special significance in the formation of square roots, it is not unreasonable to conjecture that its absence from the spectrum of an operator A is sufficient to ensure the existence of a square root of A, or, in other words, that even on not necessarily finite-dimensional Hilbert spaces, every invertible operator has a square root. (This conjecture was first called to our attention by Irving Kaplansky.) The main purpose of this paper is to prove that this conjecture is false. More precisely, we shall describe a small but interesting class of operators, derive a necessary and sufficient condition that an operator in this class have a square root, and achieve our announced purpose by exhibiting a relatively large subclass of invertible operators that do not satisfy the condition. We note in passing that our methods solve the analogous problem for nth roots, n _ 2, and that,

Communication system and technique using QR decomposition with a triangular systolic array

Abstract: An apparatus, system, and method to perform QR decomposition of an input complex matrix are described. The apparatus may include a triangular systolic array to load the input complex matrix and an identity matrix, to perform a unitary complex matrix transformation requiring three rotation angles, and to produce a complex unitary matrix and an upper triangular matrix. The upper triangular matrix may include real diagonal elements. Other embodiments are described and claimed.

An optimum iterative method for solving any linear system with a square matrix

Abstract: A method is presented to solveAx=b by computing optimum iteration parameters for Richardson's method. It requires some information on the location of the eigenvalues ofA. The algorithm yields parameters well-suited for matrices for which Chebyshev parameters are not appropriate. It therefore supplements the Manteuffel algorithm, developed for the Chebyshev case. Numerical examples are described.