The red vector is an eigenvector of the transformation while the blue one is not
In mathematics, the
transformation are important properties of this transformation. These key concepts play a major role in mathematics and, in particular, in linear algebra and functional analysis, as well as in numerous applied disciplines. The prefix eigen emphasizes the fact that these properties are important characteristics of the transformation. In many common cases knowing all eigenvalues and eigenvectors of a transformation is equivalent to the explicit knowledge of the transformation. The word eigen is German
for "own", "peculiar", or "individual": the most likely translation into English mathematical jargon would be "characteristic", and some older references do use the expressions "characteristic value", "characteristic vector" and so forth, or even "eigenwert" which is German for eigenvalue, but the more distinctive term "eigenvalue" has become standard.
sand dunes from an extinct desert. Uplift of the region started about 75 million years ago in the Laramide orogeny, a mountain-building event that is largely responsible for creating the Rocky Mountains to the east. The canyon did not start to form until 5.3 million years ago when the Gulf of California opened up and thus lowered the river's base level (its lowest point) from that of large inland lakes to sea level. About 2 million years ago volcanic activity started to deposit ash and lava over the area. The nearly 40 identified rock layers and 14 major unconformities
(gaps in the geologic record) of the Grand Canyon form one of the most studied sequences of rock in the world.
Academy Awards and won three. It is widely accepted as a classic, often cited as one of the most noteworthy films of American cinema. Deemed "culturally significant" by the United StatesLibrary of Congress in 1989, Sunset Boulevard was included in the first group of films selected for preservation in the National Film Registry
brown. Some cheeses melt smoothly, especially in the presence of acids or starch. Cheese fondue, with wine providing the acidity, is a good example of a smoothly-melted cheese dish. Other cheeses turn elastic and stringy when they melt, a quality that can be enjoyed in dishes like pizza. Some cheeses melt unevenly, their fats separating as they heat, while a few acid-curdled cheeses, including paneer and ricotta
, do not melt at all and can become firmer when cooked.
black powder rifle, the kammerlader was operated with a crank mounted on the side of the receiver
. This made it much quicker and easier to load than the weapons previously used. Kammerladers quickly gained a reputation for being fast and accurate rifles, and would have been a deadly weapon against massed ranks of infantry. The kammerladers were phased out as more modern rifles were approved for use. They were either modified for rimfire cartridges, sold off to civilians or melted for scrap. Rifles sold to civilians were often modified for use as shotguns or hunting weapons. Today it is hard to find an unmodified kammerlader, and collectors often pay high prices for them.
Arsenal's players and fans celebrate their 2004 title win
FA Premier League and is one of the most successful clubs in England. Arsenal has won thirteen First Division and Premier League titles, and ten FA Cups, as well as being the first London club to reach the UEFA Champions League final. Arsenal was founded in south-east London in 1886, but moved across the city to the Arsenal Stadium, Highbury, in 1913. In 2006, the club moved again to the new 60,000-seater Emirates Stadium in nearby Ashburton Grove. Arsenal enjoys a fierce rivalry with Tottenham Hotspur, from nearby Tottenham, whom they play in the North London derby
Korean Peninsula from the 7th to the 9th centuries. A vast number of sites from this period remain in the city today. After the kingdom fell, the city gradually declined in importance. Today Gyeongju is a typical medium-sized city, having shared in the economic, demographic, and social trends that have shaped modern South Korea. However, amidst these trends the city has retained a distinctive identity. In tourism, it is one of South Korea's most well-known destinations. In manufacturing, it profits from its proximity to major industrial centers such as Ulsan
. Gyeongju is connected to nationwide rail and expressway networks, which facilitate both industrial and tourist traffic.
Robert Arthur Lawson is considered one of New Zealand's most eminent 19th centuryarchitects. Lawson has been described as the architect who did more than any other to shape the architectural face of Victorian era cities, especially Dunedin. However, it is as the architect of over forty churches, including his monumental Gothic First Church and New Zealand's only complete "castle", Larnach Castle, that he is best remembered. Lawson is acclaimed for his work in both the Gothic revival and classical styles of architecture. He was prolific, and while isolated buildings remain in Scotland and Australia, it is in the Dunedin area that most surviving examples can now be found. Today he is held in high esteem in his adopted country; however, at the time of his death his reputation and architectural skills were still held by many in contempt following the partial collapse of his Seacliff lunatic asylum, at the time New Zealand's largest building. The great plaudits denied him in his lifetime were not to come until nearly a century after his death, when the glories of Victorian architecture began again to be recognised and appreciated.
fictional town called "Dogpatch". Dogpatch USA was a commercial success in its early years, and investors, buoyed with optimism about the park's future, decided to pursue extensive and heavily financed expansion in the form of a sister park, "Marble Falls", designed as a ski resort and convention center. But the following years would see a combination of characters and unforeseen events transform the high hopes of investors into a financial roller coaster ride which eventually ended in the park's demise. Ownership of the park changed hands many times throughout its history, and it was finally closed in 1993. Since that time much of the property of the twin parks has been neglected and frequently vandalized
, and portions of the acreage are either entangled in legal issues, in a state of redevelopment, or for sale once again.
philologist J.A.W. Bennett, it is the only prose history in English between the Conquest and the later 14th century. When William the Conqueror took England and Anglo-Norman became the official language, the Anglo-Saxon Chronicles generally ceased. The monks of Peterborough Abbey, however, continued to compile events in theirs. While the Peterborough Chronicle is not professional history, it is one of the few first-hand accounts of the period 1070 to 1154 in England written in English and from a non-courtly point of view. It is also a valuable source of information about the early Middle English
language itself. The linguistic innovations recorded in its second continuation are plentiful, and at least one innovation, the feminine pronoun "she", is first recorded in the Peterborough Chronicle.
genres, since strategy as well as dexterity are needed to complete a mission. Katamari Damacy's story, characters and settings are bizarre and heavily stylized, rarely attempting any semblance of realism. The game's simple controls and colorful, blocky graphics make it superficially appear to be targeted towards a young audience, but its quirky humor, innovative gameplay and surreal setting have attracted the attention of gamers of all ages.
An American paratrooper demonstrates removal of an S-mine
The German
Third Reich. Until production ceased with the defeat of Germany in 1945
, Germany produced over 1.93 million S-mines. These mines were responsible for inflicting heavy casualties and slowing, or even repelling, drives into German-held territory throughout the war. The design was lethal, successful and much imitated, and remains one of the definitive weapons of World War II.
jumbo jets, and computerized reservation systems. Identified by its blue globe logo and the use of "Clipper" in aircraft names and call signs, the airline was a cultural icon of the 20th century, and the unofficial flag carrier of the United States. Pan Am went through two incarnations after 1991. The second Pan Am operated from 1996 to 1998 with a focus on low-cost, long-distance flights between the U.S. and the Caribbean. The current incarnation, based in Portsmouth, New Hampshire and known as the Pan Am "Clipper Connection," is operated by Boston-Maine Airways. The airline currently flies to destinations in the northeastern United States, Florida, Dominican Republic, and Puerto Rico
miles per second), providing the first direct observation of the collision of two Solar System
objects. The collision resulted in disruptions in Jupiter's atmosphere, such as plumes and bubbles of gas, and dark spots in the atmosphere which remained visible for several months. The event was closely observed and recorded by astronomers worldwide as a result of its tremendous scientific importance, and also generated a large amount of coverage in the popular media.
Indian Rebellion of 1857 and the Second Opium War and his photographs represent the first substantial oeuvre of what came to be called photojournalism
. He had a significant impact on other photographers, and Beato's influence in Japan, where he worked with and taught numerous other photographers and artists, was particularly deep and lasting.
Montesquieu on this subject. The whole series is cited by scholars and jurists as an authoritative interpretation and explication of the meaning of the Constitution. Jurists have frequently read No. 10 to mean that the Founding Fathers did not intend the United States government to be partisan
Matthew Carey has a role as a robber of a convenience store. Directed by Greg Harrison, it was released to theatres in the United States by Sony Pictures Classics on July 22, 2005, and while its award-winning digital video photography was praised, many reviews criticised the film's story for being too ambiguous and derivative of other pictures. Critics have compared it to the work of film-makers such as David Lynch and M. Night Shyamalan
Exhibition catalog from the Semana de Arte Moderna, 1922
Week of Modern Art, the 1922 event that reshaped both literature and the visual arts
in Brazil. At the end of his life, he became the founding director of São Paulo's Department of Culture, formalizing a role he had long held as the catalyst of the city's—and the nation's—entry into artistic modernity.
"Cool" is a pop song written by American singer-songwriter Gwen Stefani and Dallas Austin for Stefani's debut solo album Love. Angel. Music. Baby. The song's musical style, and its production by Austin, was heavily inspired by music from the 1980s, and its lyrics chronicle a relationship in which two lovers have separated, but remain "cool" with each other and good friends. It received praise from critics, and parallels have been drawn by the media between the lyrical content of "Cool" and the real-life relationship Stefani had with Tony Kanal, a fellow group member of Stefani's in No Doubt. The song was released as the fourth single in the summer of 2005, and although it failed to match the chart success of its predecessor "Hollaback Girl", which reached number one in several countries, "Cool" is considered to be a worldwide hit as it reached number one in Canada, the top ten in Australia, and the top twenty in the United Kingdom and the United States, where it became Stefani's fifth top-twenty single.
theories of particle physics beyond the Standard Model. ATLAS is designed as a general-purpose experiment, so that regardless of what is produced by the collision of the accelerator's protonbeams
, the results can be measured as accurately as possible.
Queen Mary II of England, after a painting by William Wissing
Roman Catholic father, James II. Mary reigned jointly with her husband and first cousin, William III, who became the sole ruler upon her death. Popular histories usually know the joint reign as that of "William and Mary
". Mary, although a sovereign in her own right, did not wield actual power during most of her reign. She did, however, govern the realm when her husband was abroad fighting wars.
