Applications of matrices are found in most scientific fields. A major branch of numerical analysis is devoted to the development of efficient algorithms for matrix computations, a subject that is centuries old and is today an expanding area of research. In computer graphicsthey are used to manipulate 3D models and project them onto a 2-dimensional screen.
Algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matricesexpedite computations in finite element method and other computations.
In every branch of physicsincluding classical mechanicsopticselectromagnetismquantum mechanicsand quantum electrodynamicsthey are used to study physical phenomena, such as the motion of rigid bodies.
Any matrix can be multiplied element-wise by a scalar from its associated field. The rule for matrix multiplicationhowever, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second i.
In probability theory and statisticsstochastic matrices are used to describe sets of probabilities; for instance, they are used within the PageRank algorithm that ranks the pages in a Google search.
Infinite matrices occur in planetary theory and in atomic theory. A simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function. Matrix decomposition methods simplify computations, both theoretically and practically.
For example, a square matrix has an inverse if and only if its determinant is not zero. Another application of matrices is in the solution of systems of linear equations. For example, the rotation of vectors in three- dimensional space is a linear transformation, which can be represented by a rotation matrix R: If the matrix is squareit is possible to deduce some of its properties by computing its determinant.
Matrices are used in economics to describe systems of economic relationships. The product of two transformation matrices is a matrix that represents the composition of two transformations.page 1 SkillSoft program listing SkillSoft Corporation © – Information subject to change.
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Definition. A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined.
Most commonly, a matrix over a field F is a rectangular array of scalars each of which is a member of F. Most of this article focuses on real and complex matrices, that is, matrices whose .Download