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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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How do matrices behave with natural numbers?
Matrices can behave with natural numbers in various ways. Natural numbers can be used as the entries of a matrix, and basic operations such as addition, subtraction, and multiplication can be performed with natural numbers. Matrices can also be used to represent and solve systems of linear equations involving natural numbers. Additionally, natural numbers can be used to define the dimensions of a matrix, such as a 2x3 matrix, indicating that it has 2 rows and 3 columns. Overall, matrices can be used to manipulate and analyze natural numbers in a structured and organized manner.
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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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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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