Instruct: Given a question, retrieve passages that answer the question.
Query: What is a matrix in algebra?
Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, that is, the matrix sum does not depend on the order of the summands: A + B = B + A.
The transpose is compatible with addition and scalar multiplication, as expressed by ("c"A) = "c"(A) and (A + B) = A + B. Finally, (A) = A.
"Multiplication" of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an "m"-by-"n" matrix and B is an "n"-by-"p" matrix, then their "matrix product" AB is the "m"-by-"p" matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B:
In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.
Al-Karkhi (953–1029), also known as Al-Karaji, was the successor of Abū al-Wafā' al-Būzjānī (940–998) and he discovered the first numerical solution to equations of the form ax + bx = c. Al-Karkhi only considered positive roots. Al-Karkhi is also regarded as the first person to free algebra from geometrical operations and replace them with the type of arithmetic operations which are at the core of algebra today. His work on algebra and polynomials, gave the rules for arithmetic operations to manipulate polynomials. The historian of mathematics F. Woepcke, in "Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi" (Paris, 1853), praised Al-Karaji for being "the first who introduced the theory of algebraic calculus". Stemming from this, Al-Karaji investigated binomial coefficients and Pascal's triangle.
There are three main sorts of algebraic objects for which this can be done: groups, associative algebras and Lie algebras.
Ray is a one side infinite geodesic which is minimizing on each interval
Algebra was also pioneered by Persian Scientist Muhammad ibn Mūsā al-Khwārizmī during this time in his landmark text, Kitab al-Jabr wa-l-Muqabala, from which the term algebra is derived. He is thus considered to be the father of algebra. The terms algorism and algorithm are also derived from the name of al-Khwarizmi, who was responsible for introducing the Arabic numerals and Hindu-Arabic numeral system beyond the Indian Subcontinent.
Peacock's main contribution to mathematical analysis is his attempt to place algebra on a strictly logical basis. He founded what has been called the philological or symbolical school of mathematicians; to which Gregory, De Morgan and Boole belonged. His answer to Maseres and Frend was that the science of algebra consisted of two parts—arithmetical algebra and symbolical algebra—and that they erred in restricting the science to the arithmetical part. His view of arithmetical algebra is as follows: "In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs formula_1 and formula_2 denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as formula_3 we must suppose formula_4 and formula_5 to be quantities of the same kind; in others, like formula_6, we must suppose formula_4 greater than formula_5 and therefore homogeneous with it; in products and quotients, like formula_9 and formula_10 we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science."
The following structures consist of a set with a binary operation. The most common structure is that of a "group". Other structures involve weakening or strengthening the axioms for groups, and may additionally use unary operations.The main types of structures with one set having two binary operations are "rings" and "lattices". The axioms defining many of the other structures are modifications of the axioms for rings and lattices. One major difference between rings and lattices is that their two operations are related to each other in different ways. In ring-like structures, the two operations are linked by the distributive law; in lattice-like structures, the operations are linked by the absorption law.The following structures have the common feature of having two sets, "A" and "B", so that there is a binary operation from "A"×"A" into "A" and another operation from "A"×"B" into "A".Many structures here are actually hybrid structures of the previously mentioned ones.There are many examples of mathematical structures where algebraic structure exists alongside non-algebraic structure.Some algebraic structures find uses in disciplines outside of abstract algebra. The following is meant to demonstrate some specific applications in other fields.
Viète's formula for , the ratio of a circle's circumference to its diameter, is
An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a (real) inner product. A vector space equipped with such an inner product is known as a (real) inner product space. Every finite-dimensional inner product space is also a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length (or norm) of a vector, denoted , and to the angle between two vectors and by means of the formula
Exponentiation is a mathematical operation, written as , involving two numbers, the "base" and the "exponent" . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, is the product of multiplying bases:
In Rubik's cubers' parlance, a memorised sequence of moves that has a desired effect on the cube is called an algorithm. This terminology is derived from the mathematical use of "algorithm", meaning a list of well-defined instructions for performing a task from a given initial state, through well-defined successive states, to a desired end-state. Each method of solving the Rubik's Cube employs its own set of algorithms, together with descriptions of what effect the algorithm has, and when it can be used to bring the cube closer to being solved.
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Like other European languages, English contains many words derived from Arabic, often through other European languages, especially Spanish. Among them is every-day vocabulary like "sugar" ("sukkar"), "cotton" ("quṭn") or "magazine" (""). More recognizable are words like "algebra" ("al-jabr"), "alcohol" ("al-kuhūl"), "alchemy" ("al-kimiya"), "alkali", "cypher" and "zenith" (see list of English words of Arabic origin).
The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory. Von Neumann began investigating operator algebras in the 1930s, as rings of operators on a Hilbert space. The kind of algebras studied by von Neumann and his contemporaries are now known as von Neumann algebras. In the 1940s, Israel Gelfand, Mark Naimark and Irving Segal gave a definition of a kind of operator algebras called C*-algebras that on the one hand made no reference to an underlying Hilbert space, and on the other extrapolated many of the useful features of the operator algebras that had previously been studied. The spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. These techniques are now basic in abstract harmonic analysis and representation theory.
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Symmetry (from Greek συμμετρία "symmetria" "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together.
Two common classes of algebraic types are product types (i.e., tuples and records) and sum types (i.e., tagged or disjoint unions or "variant types").
Leonhard Euler considered algebraic operations on numbers modulo an integer, modular arithmetic, in his generalization of Fermat's little theorem. These investigations were taken much further by Carl Friedrich Gauss, who considered the structure of multiplicative groups of residues mod n and established many properties of cyclic and more general abelian groups that arise in this way. In his investigations of composition of binary quadratic forms, Gauss explicitly stated the associative law for the composition of forms, but like Euler before him, he seems to have been more interested in concrete results than in general theory. In 1870, Leopold Kronecker gave a definition of an abelian group in the context of ideal class groups of a number field, generalizing Gauss's work; but it appears he did not tie his definition with previous work on groups, particularly permutation groups. In 1882, considering the same question, Heinrich M. Weber realized the connection and gave a similar definition that involved the cancellation property but omitted the existence of the inverse element, which was sufficient in his context (finite groups).
Under international consensus, most braille alphabets follow the French sorting order for the 26 letters of the basic Latin alphabet, and there have been attempts at unifying the letters beyond these 26 (see international braille), though differences remain, for example in German Braille and the contractions of English Braille. This unification avoids the chaos of each nation reordering the braille code to match the sorting order of its print alphabet, as happened in Algerian Braille, where braille codes were numerically reassigned to match the order of the Arabic alphabet and bear little relation to the values used in other countries (compare modern Arabic Braille, which uses the French sorting order), and as happened in an early American version of English Braille, where the letters "w, x, y, z" were reassigned to match English alphabetical order. A convention sometimes seen for letters beyond the basic 26 is to exploit the physical symmetry of braille patterns iconically, for example, by assigning a reversed "n" to "ñ" or an inverted "s" to "sh". (See Hungarian Braille and Bharati Braille, which do this to some extent.)
In July 2009, a community-made patch was released which allowed user-made operating systems to be easily uploaded onto the TI-84 plus series. Shortly after the patch was developed, the RSA keys for the calculator's operating system were factored via the General number field sieve (GNFS) algorithm, making a software patch unnecessary. In response to this, Texas Instruments released a newer hardware revision which only accepts other, stronger RSA keys, making it harder to load user-made operating systems or older TI operating systems (2.53MP and earlier). The community has found a way around the newest limitation by discovering a way to revert to older versions of the boot code.The TI-84 Plus is based heavily on its predecessor, the TI-83 Plus. As with all other calculators in the series, the TI-84 Plus supports native Z80 assembly as well as TI's interpreted, BASIC-like language for calculators, dubbed TI-BASIC. Programming for the TI-84 Plus is nearly identical to programming for the TI-83 Plus, with a few new functions in both TI-BASIC and the calculator's assembly support that do not exist on earlier models and OS versions. Several attempts have been made at creating a C to Zilog Z80 assembly assemblers, such as SDCC.
In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.
In linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
The projection matrix has a number of useful algebraic properties. In the language of linear algebra, the projection matrix is the orthogonal projection onto the column space of the design matrix formula_26.(Note that formula_27 is the pseudoinverse of X.) Some facts of the projection matrix in this setting are summarized as follows:
The projection matrix corresponding to a linear model is symmetric and idempotent, that is, formula_33. However, this is not always the case; in locally weighted scatterplot smoothing (LOESS), for example, the hat matrix is in general neither symmetric nor idempotent.
In linear algebra, a Moore matrix, introduced by , is a matrix defined over a finite field. When it is a square matrix its determinant is called a Moore determinant (this is unrelated to the Moore determinant of a quaternionic Hermitian matrix). The Moore matrix has successive powers of the Frobenius automorphism applied to the first column, so it is an "m" × "n" matrix