This paper mainly introduces some equivalent conditions for SEP matrix, specifically by constructing some specific matrix equations and discussing whether these matrix equations have solutions in given set to determine whether a group invertible matrix is a SEP matrix.

Cited by 2Open access

Matrix Solution Method of a Grey Matrix Game

Systems evaluation, prediction and decision-making/Systems evaluation, prediction, and decision-making series · 2009

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Solution of Certain Matrix Equations

J. H. Jones · Proceedings of the American Mathematical Society · 1972

The main purpose of this paper is to obtain solutions of matrix equations of the following types, AX-XB=C, XDX+AX+XB+C=0, in which case X is an unknown n by n matrix and A,B,C, D are n by n matrices having elements belonging to the field C of complex numbers.Results obtained extend those of W. E. Roth, J. E. Potter and others concerning the existence and the representation of solutions X of the above equations.

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Product Search Solution Evaluation Matrix

Susan Aldrich · 2006

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Matrix Solution of Linear Equations

2010

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Solution of certain matrix equations

John Jones · Proceedings of the American Mathematical Society · 1972

The main purpose of this paper is to obtain solutions of matrix equations of the following types, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper X minus upper X upper B equals upper C comma upper X upper D upper X plus upper A upper X plus upper X upper B plus upper C equals 0"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>−</mml:mo> <mml:mi>X</mml:mi> <mml:mi>B</mml:mi> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mi>D</mml:mi> <mml:mi>X</mml:mi> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mi>X</mml:mi> <mml:mo>+</mml:mo> <mml:mi>X</mml:mi> <mml:mi>B</mml:mi> <mml:mo>+</mml:mo> <mml:mi>C</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">AX - XB = C,XDX + AX + XB + C = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in which case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an unknown <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrix and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A comma upper B comma upper C comma upper D"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mi>D</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A,B,C,D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrices having elements belonging to the field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of complex numbers. Results obtained extend those of W. E. Roth, J. E. Potter and others concerning the existence and the representation of solutions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the above equations.

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Matrix in Aqueous Solution by

J Un, Alicia Cochran, Michael Cima et al. · 2013

Knowledge of the oxygen levels in a tumor is a current goal of cancer research due to the importance of hypoxia on tumor growth and treatment. Previous work by the Cima group at MIT has shown that an oxygen sensor made from low molecular weight siloxanes (LMWS) in a PDMS matrix can be implanted during a biopsy and effectively measure oxygen levels in rat tumors with MRI. The Cima group also found that the sensors experienced signal loss over time when stored in air. The purpose of this investigation was to explore the loss of signal over time of the sensor when stored in aqueous solution. The signal reduction over time in environments similar to the body could be approximated by measuring the spin-lattice relaxation times of the sensor in various aqueous solutions. It was hypothesized that diffusion of the LMWS from the sensor to the surrounding environment was the potential cause of the signal loss over time, so the amount of LMWS that escaped into aqueous

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Some Basic Matrix Solution Schemes

S. Ratnajeevan H. Hoole, Yovahn Yesuraiyan R. Hoole · 2019

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Extracellular Matrix: The Networking Solution

Nancy Boudreau, Mina J. Bissell · The Cancer Handbook · 2005

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COMPUTER SOLUTION OF MATRIX PROBLEMS

Bazylevych Yuriy, Stanina Olha · International scientific and technical conference Information technologies in metallurgy and machine building · 2024

A feature of the computer solution of matrix problems is that often there is a problem of accumulation of rounding errors. This may lead to an incorrect result. J.H. Wilkinson developed efficient methods for finding eigenvalues and eigenvectors of matrices based on the well-known Francis-Kublanovska’s QR-algorithm. Now there are new problems of algebra, the methods of solution of which require further improvement. There is a problem of reducing a few initial matrices into a block-diagonal or block-triangular form. This requires the development of a new approaches to solving the problems of finding the centralizer of matrices and constructing an algebra with a unit generated by these matrices. For the first of these problems, it was possible to create an effective method. The next problem is to create an efficient algorithm for constructing the algebra generated by matrices

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