In mathematics, a block matrix or a partitioned matrix is a matrix, interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned; this notion can be made more precise for an n by m matrix M by partitioning n into a collection r o w g r o u p s, partitioning m into a collection c o l g r o u p s. The original matrix is considered as the "total" of these groups, in the sense that the entry of the original matrix corresponds in a 1-to-1 way with some offset entry of some, where x ∈ r o w g r o u p s and y ∈ c o l g r o u p s. Block matrix algebra arises in general from biproducts in categories of matrices; the matrix P = can be partitioned into four 2×2 blocks P 11 =, P 12 =, P 21 =, P 22 =.
The partitioned matrix can be written as P =. It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors; the partitioning of the factors is not arbitrary and requires "conformable partitions" between two matrices A and B such that all submatrix products that will be used are defined. Given an matrix A with q row partitions and s column partitions A = and a matrix B with s row partitions and r column partitions B = [ B 11 B 12 ⋯ B 1
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Indicated by the Greek letter sigma, they are denoted by tau when used in connection with isospin symmetries, they are σ 1 = σ x = σ 2 = σ y = σ 3 = σ z =. These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field; each Pauli matrix is Hermitian, together with the identity matrix I, the Pauli matrices form a basis for the vector space of 2 × 2 Hermitian matrices. Hermitian operators represent observables, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space. In the context of Pauli's work, σk represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space ℝ3; the Pauli matrices generate transformations in the sense of Lie algebras: the matrices iσ1, iσ2, iσ3 form a basis for s u, which exponentiates to the special unitary group SU.
The algebra generated by the three matrices σ1, σ2, σ3 is isomorphic to the Clifford algebra of ℝ3. All three of the Pauli matrices can be compacted into a single expression: σ a = where i = √−1 is the imaginary unit, δab is the Kronecker delta, which equals +1 if a = b and 0 otherwise; this expression is useful for "selecting" any one of the matrices numerically by substituting values of a = 1, 2, 3, in turn useful when any of the matrices is to be used in algebraic manipulations. The matrices are involutory: σ 1 2 = σ 2 2 = σ 3 2 = − i σ 1 σ 2 σ 3 = = I where I is the identity matrix; the determinants and traces of the Pauli matrices are: det σ i = − 1, tr σ i = 0. From which, we can deduce that the eigenvalues of each σi are ±1. With the inclusion of the identity matrix, I, the Pauli matrices form an orthogonal basis of the real Hilbert space of 2 × 2 complex Hermitian matrices, H 2, the complex Hilbert space of all 2 × 2 matrices, M 2, 2; each of the Pauli matrices has two eigenvalues, +1 and −1.
The corresponding normalized eigenvectors are: ψ x + = 1 2, ψ x − = 1 2 ( 1 − 1
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. With this distance, Euclidean space becomes a metric space; the associated norm is called the Euclidean norm. Older literature refers to the metric as the Pythagorean metric. A generalized term for the Euclidean norm is the L2 L2 distance; the Euclidean distance between points p and q is the length of the line segment connecting them. In Cartesian coordinates, if p = and q = are two points in Euclidean n-space the distance from p to q, or from q to p is given by the Pythagorean formula: The position of a point in a Euclidean n-space is a Euclidean vector. So, p and q may be represented as Euclidean vectors, starting from the origin of the space with their tips ending at the two points; the Euclidean norm, or Euclidean length, or magnitude of a vector measures the length of the vector: ‖ p ‖ = p 1 2 + p 2 2 + ⋯ + p n 2 = p ⋅ p, where the last expression involves the dot product.
Describing a vector as a directed line segment from the origin of the Euclidean space, to a point in that space, its length is the distance from its tail to its tip. The Euclidean norm of a vector is seen to be just the Euclidean distance between its tail and its tip; the relationship between points p and q may involve a direction, so when it does, this relationship can itself be represented by a vector, given by q − p =. In a two- or three-dimensional space, this can be visually represented as an arrow from p to q. In any space it can be regarded as the position of q relative to p, it may be called a displacement vector if p and q represent two positions of some moving point. The Euclidean distance between p and q is just the Euclidean length of this displacement vector:, equivalent to equation 1, to: ‖ q − p ‖ = ‖ p ‖ 2 + ‖ q ‖ 2 − 2 p ⋅ q. In the context of Euclidean geometry, a metric is established in one dimension by fixing two points on a line, choosing one to be the origin; the length of the line segment between these points defines the unit of distance and the direction from the origin to the second point is defined as the positive direction.
This line segment may be translated along the line to build longer segments whose lengths correspond to multiples of the unit distance. In this manner real numbers can be associated to points on the line and these are the Cartesian coordinates of the points on what may now be called the real line; as an alternate way to establish the metric, instead of choosing two points on the line, choose one point to be the origin, a unit of length and a direction along the line to call positive. The second point is uniquely determined as the point on the line, at a distance of one positive unit from the origin; the distance between any two points on the real line is the absolute value of the numerical difference of their coordinates. It is common to identify the name of a point with its Cartesian coordinate, thus if p and q are two points on the real line the distance between them is given by: 2 = | q − p |. In one dimension, there is a single homogeneous, translation-invariant metric, up to a scale factor of length, the Euclidean distance.
In higher dimensions there are other possible norms. In the Euclidean plane, if p = and q = the distance is given by d = 2 + 2; this is equivalent to the Pythagorean theorem. Alternatively, it follows from that if the polar coordinates of the point p are and those of q are the distance between the points is r 1 2