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Eigenvalues
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigendecomposition Of A Matrix
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Fundamental theory of matrix eigenvectors and eigenvalues A (nonzero) vector of dimension is an eigenvector of a square matrix if it satisfies a linear equation of the form \mathbf \mathbf = \lambda \mathbf for some scalar . Then is called the eigenvalue corresponding to . Geometrically speaking, the eigenvectors of are the vectors that merely elongates or shrinks, and the amount that they elongate/shrink by is the eigenvalue. The above equation is called the eigenvalue equation or the eigenvalue problem. This yields an equation for the eigenvalues p\left(\lambda\right) = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Greek
Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic or Homeric period (), and the Classical period (). Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers. It has contributed many words to English vocabulary and has been a standard subject of study in educational institutions of the Western world since the Renaissance. This article primarily contains information about the Epic and Classical periods of the language, which are the best-attested periods and considered most typical of Ancient Greek. From the Hellenistic period (), Ancient Greek was followed by Koine Greek, which is regarded as a separate historical stage, though its earliest form closely resembles Attic Greek, and its latest form approaches Medieval Greek, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modern Greek
Modern Greek (, or , ), generally referred to by speakers simply as Greek (, ), refers collectively to the dialects of the Greek language spoken in the modern era, including the official standardized form of the language sometimes referred to as Varieties of Modern Greek#Standard Modern Greek, Standard Modern Greek. The end of the Medieval Greek period and the beginning of Modern Greek is often symbolically assigned to the fall of the Byzantine Empire in 1453, even though that date marks no clear linguistic boundary and many characteristic features of the modern language arose centuries earlier, having begun around the fourth century AD. During most of the Modern Greek period, the language existed in a situation of diglossia, with regional spoken dialects existing side by side with learned, more archaic written forms, as with the vernacular and learned varieties (''Dimotiki'' and ''Katharevousa'') that co-existed in Greece throughout much of the 19th and 20th centuries. Variet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Alphabet
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC. It was derived from the earlier Phoenician alphabet, and is the earliest known alphabetic script to systematically write vowels as well as consonants. In Archaic Greece, Archaic and early Classical Greece, Classical times, the Greek alphabet existed in Archaic Greek alphabets, many local variants, but, by the end of the 4th century BC, the Ionia, Ionic-based Euclidean alphabet, with 24 letters, ordered from alpha to omega, had become standard throughout the Greek-speaking world and is the version that is still used for Greek writing today. The letter case, uppercase and lowercase forms of the 24 letters are: : , , , , , , , , , , , , , , , , , , , , , , , The Greek alphabet is the ancestor of several scripts, such as the Latin script, Latin, Gothic alphabet, Gothic, Coptic script, Coptic, and Cyrillic scripts. Throughout antiquity, Greek had only a single uppercas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of The Greek Alphabet
The history of the Greek alphabet starts with the adoption of Phoenician letter forms in the 9th–8th centuries BC during early Archaic Greece and continues to the present day. The Greek alphabet was developed during the Iron Age, centuries after the loss of Linear B, the syllabic script that was used for writing Mycenaean Greek until the Late Bronze Age collapse and Greek Dark Age. This article concentrates on the development of the alphabet before the modern codification of the standard Greek alphabet. The Phoenician alphabet was consistently explicit only about consonants, though even by the 9th century BC it had developed '' matres lectionis'' to indicate some, mostly final, vowels. This arrangement is much less suitable for Greek than for Semitic languages, and these ''matres lectionis'', as well as several Phoenician letters which represented consonants not present in Greek, were adapted according to the acrophonic principle to represent Greek vowels consistently, if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First-order Logic
First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all humans are mortal", in first-order logic one can have expressions in the form "for all ''x'', if ''x'' is a human, then ''x'' is mortal", where "for all ''x"'' is a quantifier, ''x'' is a variable, and "... ''is a human''" and "... ''is mortal''" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A theory about a topic, such as set theory, a theory for groups,A. Tarski, ''Undecidable Theories'' (1953), p. 77. Studies in Logic and the Foundation of Mathematics, North-Holland or a formal theory o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal Matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is \left begin 3 & 0 \\ 0 & 2 \end\right/math>, while an example of a 3×3 diagonal matrix is \left begin 6 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 4 \end\right/math>. An identity matrix of any size, or any multiple of it is a diagonal matrix called a ''scalar matrix'', for example, \left begin 0.5 & 0 \\ 0 & 0.5 \end\right/math>. In geometry, a diagonal matrix may be used as a '' scaling matrix'', since matrix multiplication with it results in changing scale (size) and possibly also shape; only a scalar matrix results in uniform change in scale. Definition As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix with columns and rows is diagonal if \forall i,j \in \, i \ne j \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epichoric Alphabets
Many local variants of the Greek alphabet were employed in ancient Greece during the Archaic Greece, archaic and Classical Greece, early classical periods, until around 400 BC, when they were replaced by the classical 24-letter alphabet that is the standard today. All forms of the Greek alphabet were originally based on the shared inventory of the 22 symbols of the Phoenician alphabet, with the exception of the letter Samekh, whose Greek counterpart Xi (letter), Xi () was used only in a subgroup of Greek alphabets, and with the common addition of Upsilon () for the vowel . The local, so-called ''epichoric'', alphabets differed in many ways: in the use of the consonant symbols , and ; in the use of the innovative long vowel letters ( and ), in the absence or presence of Η in its original consonant function (); in the use or non-use of certain archaic letters ( = , = , = ); and in many details of the individual shapes of each letter. The system now familiar as the standa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
El (Cyrillic)
El (Л л or Ʌ ʌ; italics: ) is a letter of the Cyrillic script. El commonly represents the alveolar lateral approximant . In Slavic languages it may be either palatalized or slightly velarized; see below. History The Cyrillic letter El was derived from the Greek letter lambda (Λ λ). In the Early Cyrillic alphabet its name was (''ljudije''), meaning "people". In the Cyrillic numeral system, Л had a value of 30. Forms El has two forms: one form resembles Greek capital Lambda (Ʌ ʌ), and the other form resembles the Hebrew letter ת (Л л). In some typeface A typeface (or font family) is a design of Letter (alphabet), letters, Numerical digit, numbers and other symbols, to be used in printing or for electronic display. Most typefaces include variations in size (e.g., 24 point), weight (e.g., light, ...s the Cyrillic letter El has a grapheme which may be confused with the Cyrillic letter Pe (П п). Note that Pe has a straight left leg, without the hoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Numerals
Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, is a numeral system, system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal number (linguistics), ordinal numbers and in contexts similar to those in which Roman numerals are still used in the Western world. For ordinary cardinal number (linguistics), cardinal numbers, however, modern Greece uses Arabic numerals. History The Minoans, Minoan and Mycenaean civilizations' Linear A and Linear B alphabets used a different system, called Aegean numerals, which included number-only symbols for powers of ten: = 1, = 10, = 100, = 1000, and = 10000. Attic numerals composed another system that came into use perhaps in the 7th century BC. They were acrophonic, derived (after the initial one) from the first letters of the names of the numbers represented. They ran = 1, = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann's Hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its Root of a function, zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important List of unsolved problems in mathematics, unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of Hilbert's problems, twenty-three unsolved problems; it is also one of the Millennium Prize Problems of the Clay Mathematics Institute, which offers US$1 million for a solution to any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann zeta function ''ζ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
