- Introduction to Matrices | Boundless Algebra
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- Understanding Math - Introduction to Matrices
And there is a relationship between the movie, which is about a virtual reality constructed by super-smart computers, and the notion of what a matrix is when you study it in mathematics, or when you study it in computer science. And the connection really is that matrices are used a lot when you are simulating things or when you're constructing things in computer science, especially in, frankly, computer graphics.
So the super-intelligent robots that made the matrix in the movie Matrix were probably using matrices in order to do it, if they actually did exist. Now, what is a matrix then? Well, that's a fairly simple answer. It's just a rectangular array of numbers. So for example, this right over here.
If I have 1, 0, negative 7, pi, 5, and-- I don't know-- 11, this is a matrix. This is a matrix where 1, 0, negative 7, pi-- each of those are an entry in the matrix. This matrix right over here has two rows. And it has three columns. And because it has two rows and three columns, people will often say that this is a 2 by 3 matrix.
Whenever they say it's something by something matrix, they're telling you that it has two rows-- so you see the two rows right over there. And they are telling you that it has three columns. You see the three columns right over there. I could give you other examples of a matrix. So I could have a 1 by 1 matrix. So I could have the matrix 1. This right over here is a 1 by 1 matrix. It has one row, one column. I could have a matrix like this-- 3, 7, and What is this? We'll see in the next section how to evaluate this determinant. It has value Determinants - derived from a square matrix, a determinant needs to be multiplied out to give a single number.
Large Determinants - this section will help you to understand smaller determinants. Multiplication of Matrices - how to multiply matrices of different sizes.
Introduction to Matrices | Boundless Algebra
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An empty matrix is a matrix in which the number of rows or columns or both is zero. For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them. The determinant of the 0-by-0 matrix is 1 as follows from regarding the empty product occurring in the Leibniz formula for the determinant as 1.
There are numerous applications of matrices, both in mathematics and other sciences. Some of them merely take advantage of the compact representation of a set of numbers in a matrix. For example, in game theory and economics , the payoff matrix encodes the payoff for two players, depending on which out of a given finite set of alternatives the players choose. For example, 2-by-2 rotation matrices represent the multiplication with some complex number of absolute value 1, as above. A similar interpretation is possible for quaternions  and Clifford algebras in general.
Early encryption techniques such as the Hill cipher also used matrices. However, due to the linear nature of matrices, these codes are comparatively easy to break. Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy. Examples are the overlap matrix and the Fock matrix used in solving the Roothaan equations to obtain the molecular orbitals of the Hartree—Fock method. The adjacency matrix of a finite graph is a basic notion of graph theory. Matrices containing just two different values 1 and 0 meaning for example "yes" and "no", respectively are called logical matrices.
The distance or cost matrix contains information about distances of the edges. Therefore, specifically tailored matrix algorithms can be used in network theory. Quadratic programming can be used to find global minima or maxima of quadratic functions closely related to the ones attached to matrices see above. Partial differential equations can be classified by considering the matrix of coefficients of the highest-order differential operators of the equation.
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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. 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. Stochastic matrices are square matrices whose rows are probability vectors , that is, whose entries are non-negative and sum up to one. Stochastic matrices are used to define Markov chains with finitely many states. Properties of the Markov chain like absorbing states , that is, states that any particle attains eventually, can be read off the eigenvectors of the transition matrices.
Statistics also makes use of matrices in many different forms. The covariance matrix encodes the mutual variance of several random variables. Random matrices are matrices whose entries are random numbers, subject to suitable probability distributions , such as matrix normal distribution. Beyond probability theory, they are applied in domains ranging from number theory to physics.
Linear transformations and the associated symmetries play a key role in modern physics.
For example, elementary particles in quantum field theory are classified as representations of the Lorentz group of special relativity and, more specifically, by their behavior under the spin group. Concrete representations involving the Pauli matrices and more general gamma matrices are an integral part of the physical description of fermions , which behave as spinors.
The Cabibbo—Kobayashi—Maskawa matrix , in turn, expresses the fact that the basic quark states that are important for weak interactions are not the same as, but linearly related to the basic quark states that define particles with specific and distinct masses. The first model of quantum mechanics Heisenberg , represented the theory's operators by infinite-dimensional matrices acting on quantum states.
One particular example is the density matrix that characterizes the "mixed" state of a quantum system as a linear combination of elementary, "pure" eigenstates. Another matrix serves as a key tool for describing the scattering experiments that form the cornerstone of experimental particle physics: Collision reactions such as occur in particle accelerators , where non-interacting particles head towards each other and collide in a small interaction zone, with a new set of non-interacting particles as the result, 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.
A general application of matrices in physics is to the description of linearly coupled harmonic systems. The equations of motion of such systems can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement vector to characterize the interactions.
The best way to obtain solutions is to determine the system's eigenvectors , its normal modes , by diagonalizing the matrix equation. Techniques like this are crucial when it comes to the internal dynamics of molecules : the internal vibrations of systems consisting of mutually bound component atoms.
Geometrical optics provides further matrix applications. In this approximative theory, the wave nature of light is neglected. The result is a model in which light rays are indeed 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 multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix : the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element.
Actually, there are two kinds of matrices, viz. Traditional mesh analysis and nodal analysis in electronics lead to a system of linear equations that can be described with a matrix. The behaviour of many electronic components can be described using matrices. Let A be a 2-dimensional vector with the component's input voltage v 1 and input current i 1 as its elements, and let B be a 2-dimensional vector with the component's output voltage v 2 and output current i 2 as its elements.
Calculating a circuit now reduces to multiplying matrices. Matrices have a long history of application in solving linear equations but they were known as arrays until the s. The Chinese text The Nine Chapters on the Mathematical Art written in 10th—2nd century BCE is the first example of the use of array methods to solve simultaneous equations ,  including the concept of determinants. The term "matrix" Latin for "womb", derived from mater —mother  was coined by James Joseph Sylvester in ,  who understood a matrix as an object giving rise to a number of determinants today called minors , that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows.
In an paper, Sylvester explains:. Arthur Cayley published a treatise on geometric transformations using matrices that were not rotated versions of the coefficients being investigated as had previously been done. Instead he defined operations such as addition, subtraction, multiplication, and division as transformations of those matrices and showed the associative and distributive properties held true.
Cayley investigated and demonstrated the non-commutative property of matrix multiplication as well as the commutative property of matrix addition. He was instrumental in proposing a matrix concept independent of equation systems. In Cayley published his A memoir on the theory of matrices   in which he proposed and demonstrated the Cayley—Hamilton theorem. The modern study of determinants sprang from several sources. Eisenstein further developed these notions, including the remark that, in modern parlance, matrix products are non-commutative.
He also showed, in , that the eigenvalues of symmetric matrices are real. At that point, determinants were firmly established. Frobenius , working on bilinear forms , generalized the theorem to all dimensions Also at the end of the 19th century the Gauss—Jordan elimination generalizing a special case now known as Gauss elimination was established by Jordan.
In the early 20th century, matrices attained a central role in linear algebra,  partially due to their use in classification of the hypercomplex number systems of the previous century. The inception of matrix mechanics by Heisenberg , Born and Jordan led to studying matrices with infinitely many rows and columns. Bertrand Russell and Alfred North Whitehead in their Principia Mathematica — use the word "matrix" in the context of their axiom of reducibility. They proposed this axiom as a means to reduce any function to one of lower type, successively, so that at the "bottom" 0 order the function is identical to its extension :.
Alfred Tarski in his Introduction to Logic used the word "matrix" synonymously with the notion of truth table as used in mathematical logic. From Wikipedia, the free encyclopedia. Two-dimensional array of numbers with specific operations. For other uses, see Matrix. For the physics topic, see Matrix string theory.
Main articles: Matrix addition , Scalar multiplication , and Transpose. Main article: Matrix multiplication. Main article: Row operations. Main articles: Linear equation and System of linear equations. Main articles: Linear transformation and Transformation matrix. Main article: Square matrix. Main article: Identity matrix. Main article: Orthogonal matrix. Main article: Determinant.
Main article: Eigenvalues and eigenvectors. Main articles: Matrix decomposition , Matrix diagonalization , Gaussian elimination , and Montante's method. Main article: Matrix group. Further information: Symmetry in physics.
Understanding Math - Introduction to Matrices
Mathematics portal. Laurie Rosatone. Bryan and T. SIAM Review, 48 3 —, TED ED. Retrieved April 6, See also stiffness method.