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Nonnegative Matrices and Applications
This book provides an integrated treatment of the theory of nonnegative matrices (matrices with only positive numbers or zero as entries) and some related classes of positive matrices, concentrating on connections with game theory, combinatorics, inequalities, optimisation and mathematical economics.The wide variety of applications, which include price fixing, scheduling and the fair division problem, have been carefully chosen both for their elegant mathematical content and for their accessibility to students with minimal preparation.Many results in matrix theory are also presented. The treatment is rigorous and almost all results are proved completely.These results and applications will be of great interest to researchers in linear programming, statistics and operations research.The minimal prerequisites also make the book accessible to first-year graduate students.
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Positivity : Confidence, Resilience, Motivation
Do you want more optimism, confidence, resilience and motivation?Then this book is for you! We currently live in a time of unprecedented challenges, uncertainty, overwhelming stress, loss of hope at times and a need for mental strength and adaptability to a new way of life. Traditional 'positive thinking', trying to constantly tell yourself that everything is OK, no matter the circumstances, just isn't enough to make any lasting difference. After 30 years working as a therapist, Paul McKenna has developed a unique approach – one that can guide you towards an all-round feeling of positivity – putting you into optimal states of mind, building up your resilience and enabling good decisions and actions that lead to successful results in life. There is now irrefutable scientific evidence that shows that particular ways of thinking and acting produce tangible positive results in people's ability to deal with challenges and their overall quality of life.The research also shows that this mindset can be learned quickly just like any habit. And that is why this book is a practical psychological system in how to survive and thrive – how to discover your own natural most powerful resources for self-care, self-belief and for taking control of your life.
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Can matrices be divided?
Matrices cannot be divided in the same way that numbers can be divided. However, matrices can be multiplied by the inverse of another matrix, which is similar to division. This operation is used to solve systems of linear equations and find solutions to matrix equations. Overall, while matrices cannot be divided in the traditional sense, there are operations that can achieve similar results.
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What are invertible matrices?
Invertible matrices are square matrices that have an inverse, meaning that there exists another matrix that, when multiplied with the original matrix, results in the identity matrix. The inverse of a matrix A is denoted as A^-1, and it satisfies the property that A * A^-1 = A^-1 * A = I, where I is the identity matrix. Invertible matrices are also called nonsingular matrices, and they are important in various areas of mathematics and applications, such as solving systems of linear equations and in transformations in linear algebra.
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Which matrices contain symbols?
Matrices that contain symbols are typically referred to as symbolic matrices. These matrices can contain variables, parameters, or other mathematical symbols instead of specific numerical values. Symbolic matrices are commonly used in various fields of mathematics, such as linear algebra, calculus, and differential equations, where the exact values of the elements are not known or need to be represented in a general form. These matrices are useful for performing algebraic manipulations, solving equations, and representing mathematical relationships in a more general and abstract manner.
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What are representation matrices?
Representation matrices are matrices that represent linear transformations or operators. They are used to represent the action of a linear transformation on a vector space. The elements of the representation matrix correspond to the coefficients of the linear combination of the basis vectors of the vector space. By using representation matrices, we can easily perform operations such as composition of linear transformations and finding the inverse of a linear transformation.
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Direct Methods for Sparse Matrices
The subject of sparse matrices has its root in such diverse fields as management science, power systems analysis, surveying, circuit theory, and structural analysis.Efficient use of sparsity is a key to solving large problems in many fields.This second edition is a complete rewrite of the first edition published 30 years ago.Much has changed since that time. Problems have grown greatly in size and complexity; nearly all examples in the first edition were of order less than 5,000 in the first edition, and are often more than a million in the second edition.Computer architectures are now much more complex, requiring new ways of adapting algorithms to parallel environments with memory hierarchies. Because the area is such an important one to all of computational science and engineering, a huge amount of research has been done in the last 30 years, some of it by the authors themselves.This new research is integrated into the text with a clear explanation of the underlying mathematics and algorithms. New research that is described includes new techniques for scaling and error control, new orderings, new combinatorial techniques for partitioning both symmetric and unsymmetric problems, and a detailed description of the multifrontal approach to solving systems that was pioneered by the research of the authors and colleagues.This includes a discussion of techniques for exploiting parallel architectures and new work for indefinite and unsymmetric systems.
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Introduction to Matrices & Linear Transformations
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Which matrices are commutative?
Two matrices are commutative if their product is the same regardless of the order in which they are multiplied. In other words, for matrices A and B, if A*B = B*A, then they are commutative. However, not all matrices are commutative. In general, matrices are commutative only if they are scalar multiples of the identity matrix, or if they are diagonal matrices with distinct diagonal entries.
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What are stochastic matrices?
Stochastic matrices are square matrices in which each element represents the probability of transitioning from one state to another in a stochastic process. The elements of a stochastic matrix are non-negative and each row of the matrix sums to 1, representing the probabilities of transitioning to all possible states from a given state. Stochastic matrices are commonly used in the study of Markov chains, where they describe the probabilities of transitioning between different states over time. These matrices are important in modeling various real-world phenomena such as population dynamics, financial markets, and biological systems.
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What are 2x2 diagonal matrices?
A 2x2 diagonal matrix is a square matrix with only diagonal elements, while all other elements are zero. In other words, it is a matrix where the entries outside the main diagonal are all zero. For example, a 2x2 diagonal matrix may look like: [ a 0 ] [ 0 b ] where 'a' and 'b' are the diagonal elements. These matrices are commonly used in various mathematical operations and applications, such as in linear algebra and physics.
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What are matrices used for?
Matrices are used in various fields such as mathematics, physics, computer science, and engineering to represent and solve systems of linear equations, perform transformations, and analyze data. They are also used in computer graphics to represent transformations of objects in 3D space. Matrices are essential in solving problems involving multiple variables and constraints, making them a powerful tool in various applications.
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