A Study of The Applications of Matrices and R^(n) Projections By Corey Messonnier

1. A Study of The Applications of Matrices and R^(n) ProjectionsBy Corey Messonnier

2. Matrices and R^(n) projections Matrices are rectangular arrays of numbers, symbols, or expressions, where the individual entries are called its elements. An R^(n) projection is a mapping from an n dimensional space to another space of n or m dimensions.

3. Some examples of matrixes and R^(n) projection in practical applications • Graph theory: b) Symmetries and transformations in physics: c) Linear combinations of quantum states: d) Normal modes: e) Geometrical optics f) Electronics:

4. some examples of matrixes and R^(n) projection in mathematical applications • Analysis and geometry: b) Probability theory and statistics: c) Representations of equations:

5. Graph Theory: 1. is the theory of an adjacency matrix of a finite graph, in which the matrix saves which vertices of the graph that are connected by edges. 2. The concepts can be applied to websites connecting to hyperlinks or cities connected by roads. In these cases, the matrices are usually sparse matrices which are matrices containing few nonzero entries.

6. Symmetries and transformations in physics Examples Are some elementary particles in quantum field theory are classified as representations of the Lorentz group of special relativity and also by their behavior under the spin group. Another example is quarks: with the three lightest quarks, there is a group-theoretical representation involving the special unitary group SU(3); for their calculations, physicists use a convenient matrix representation known as the Gell-Mann matrices. The Gell-Mann matrices are also used for the SU(3) gauge group that forms the basis of the modern description of strong nuclear interactions, called quantum chromodynamics. The Cabibbo-Kobayashi-Maskawa matrix, is an expression of the basic quarks states that are important for weak interactions that are not the same as, but linearly related to, the basic quarks states that define particles with specific and distinct masses.

7. Linear combinations of quantum states • The 1st model of quantum mechanics was representing the theory's operators by infinite-dimensional matrices acting on quantum states.  This area of study is also referred to as matrix mechanics. One particular example is the density matrix that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" eigenstates. • Collision reactions such as those that occur in particle accelerators are where non-interacting particles head towards each other and collide in a small interaction zone. The result of these types of collision reactions is the production of a new set of non-interacting particles, which can be described as the scalar product of outgoing particle states and a linear combination of ingoing particle states. The linear combination is given by a matrix known as the S-matrix, which encodes all information about the possible interactions between particles.

8. Normal modes • Harmonic systems • Equations of Motion 3. Uses of eigenvectors in normal modes

9. Geometrical optics The wave nature of light can be modeled with matrices in which light rays are represented as geometrical rays. If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as the multiplication of a two-component vectors with a two-by-two matrix called a ray transfer matrix. The vector components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. There are two kinds of matrices: (1) a refraction matrix describing the refraction at a lens surface; (2) and a translation matrix, describing the translation of the plane of reference to the next refracting surface, where another refraction matrix is applied. The optical system, consisting of a combination of lenses and/or reflective elements, is simply described by the matrix resulting from the product of the component matrices.

10. Electronics 1. Mesh analysis 2. Electronic components

11. Analysis and geometry • The Hessian matrix is a matrix of a differentiable function; ƒ: Rn → R, which consists of the second derivatives of ƒ with respect to the several coordinate directions. That is, it encodes information about the local growth behavior of the function: given a critical point x = (x1, ..., xn), i.e., a point where the first partial derivatives of ƒ vanish, the function has a local minimum if the Hessian matrix is positive definite.  • Jacobi matrix The Jacobi matrix is also another good example in which a differentiable map f: Rn → Rm. If we let f1, ..., fm denote the components of f, then the Jacobi matrix is defined as if n > m, and if the rank of the Jacobi matrix attains its maximal value m, f is locally invertible at that point, by the implicit function theorem. This theorem is a tool that allows relations to be converted to functions. 3. Partial differential equations Partial differential equations can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation. For elliptic partial differential equations this matrix is positive definite, which has decisive influence on the set of possible solutions of the equation in question. 4. The finite element method is an important numerical method to solve partial differential equations, widely applied in simulating complex physical systems. It attempts to approximate the solution to some equation by piecewise linear functions, where the pieces are chosen with respect to a sufficiently fine grid, which in turn can be recast as a matrix equation.

12. Probability theory and statistics 1.Stochastic matrices 2. Random matrices

13. Representations of equations • Augmented matrices • complex numbers can be show in real 2 x 2 matrices under which addition and multiplication of complex numbers and matrices correspond to each other. 3. There are at least two ways of representing the quaternions as matrices in such a way that the quaternion addition and multiplication correspond to matrix addition and matrix multiplication. One way is to use 2×2 complex matrices, and the other is to use 4×4 real matrices. In each case, the representation given is one of a family of linearly related representations. In the terminology of abstract algebra, these are injective homomorphisms from H to the matrix rings M2(C) and M4(R).

14. Work cited • http://en.wikipedia.org/wiki/Matrix_(mathematics) • Notes from matrix analysis