In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or by I if the size is immaterial or can be trivially determined by the context. Less some mathematics books use U or E to represent the identity matrix, meaning "unit matrix" and the German word Einheitsmatrix respectively. I 1 =, I 2 =, I 3 =, ⋯, I n =; when A is m×n, it is a property of matrix multiplication that I m A = A I n = A. In particular, the identity matrix serves as the unit of the ring of all n×n matrices and as the identity element of the general linear group GL consisting of all invertible n×n matrices. Where n×n matrices are used to represent linear transformations from an n-dimensional vector space to itself, In represents the identity function, regardless of the basis; the ith column of an identity matrix is the unit vector ei. It follows that the determinant of the identity matrix is 1, the trace is n.
Using the notation, sometimes used to concisely describe diagonal matrices, we can write I n = diag . It can be written using the Kronecker delta notation: i j = δ i j; the identity matrix has the property that, when it is the product of two square matrices, the matrices can be said to be the inverse of one another. The identity matrix is the only idempotent matrix with non-zero determinant; that is, it is the only matrix such that when multiplied by itself, the result is itself. The principal square root of an identity matrix is itself, this is its only positive-definite square root. However, every identity matrix with at least two rows and columns has an infinitude of symmetric square roots. Binary matrix Zero matrix Unitary matrix Matrix of ones Square root of a 2 by 2 identity matrix Pauli matrices "Identity matrix". PlanetMath
The Hansa-Brandenburg W.11 was a fighter floatplane built in Germany in 1917 as a more powerful version of the KDW. Similar in general configuration to its predecessor, the W.11 shared the same unusual interplane strut arrangement, featured fins above and below the fuselage. Only three examples were built. General characteristics Crew: One pilot Length: 8.10 m Wingspan: 10.10 m Height: 3.32 m Wing area: 31.4 m2 Empty weight: 935 kg Gross weight: 1,233 kg Powerplant: 1 × Benz Bz. IVa, 160 kW Performance Maximum speed: 176 km/h Range: 350 km Armament 2 × fixed, forward-firing 7.92 mm MG 08 machine guns Taylor, Michael J. H.. Jane's Encyclopedia of Aviation. London: Studio Editions. P. 472. World Aircraft Information Files. London: Bright Star Publishing. Pp. File 896 Sheet 12
John Belding was an early settler of Norwalk, Connecticut. He was a member of the General Assembly of the Colony of Connecticut from Norwalk in the sessions of October 1691 and May 1705, he was most the son of William Belding and Thomasine Sherwood, although at least one record shows his father as John Belding of Wethersfield. He was the brother of the early settler of Deerfield, Massachusetts, he is recorded as living in Norwalk as early as 1673. On April 30, 1690, he was appointed to a committee to fortify the meeting house. On January 16, 1694, he was appointed to a committee to replace the deceased Reverend Thomas Hanford as minister for the town. In 1708, he was one of the purchasers of Ridgefield, along with Matthew Seymour, Matthias St. John, Samuel Keeler, he died in 1713, his widow, Ruth married John Copp, the town clerk. Grandfather of Thomas Belden, member of the Connecticut House of Representatives
Lakewood is a city in Pierce County, United States. The population was 58,163 at the 2010 census. Lakewood was incorporated on February 28, 1996. Historical names include Lakes District. Lakewood is the second-largest city in Pierce County and is home to the Clover Park School District, the Lakewood Water District, Fort Steilacoom Park and Western State Hospital, a regional state psychiatric hospital. Thornewood Castle was built in the vicinity of Tacoma in the area, now Lakewood. Lakewood is located at 47°10′N 122°32′W. According to the United States Census Bureau, the city has a total area of 18.95 square miles, of which, 17.17 square miles is land and 1.78 square miles is water. There are several lakes within the city limits. A number of small creeks flow through Lakewood; the largest of these, Chambers Creek, flows from Lake Steilacoom to Chambers Bay between nearby University Place and Steilacoom. The western terminus of State Route 512 is in Lakewood, at its intersection with Interstate 5; the proposed State Route 704 would be the only other state highway entering the city.
In 2004, the City of Lakewood formed the community's first Police Department, a move that has dropped the crime in recent years and has created a safer community in general. As of the census of 2010, there were 58,163 people, 24,069 households, 14,412 families living in the city; the population density was 3,387.5 inhabitants per square mile. There were 26,548 housing units at an average density of 1,546.2 per square mile. The racial makeup of the city was 59.3% White, 11.8% African American, 1.3% Native American, 9.0% Asian, 2.6% Pacific Islander, 7.3% from other races, 8.7% from two or more races. Hispanic or Latino of any race were 15.3% of the population. There were 24,069 households of which 29.8% had children under the age of 18 living with them, 39.3% were married couples living together, 15.1% had a female householder with no husband present, 5.5% had a male householder with no wife present, 40.1% were non-families. 32.3% of all households were made up of individuals and 9.5% had someone living alone, 65 years of age or older.
The average household size was 2.36 and the average family size was 2.98. The median age in the city was 36.6 years. 22.7% of residents were under the age of 18. The gender makeup of the city was 51.0 % female. As of the census of 2000, there were 58,211 people, 23,792 households, 15,084 families living in the city; the population density was 3,401.3 people per square mile. There were 25,396 housing units at an average density of 1,483.9 per square mile. The racial makeup of the city was 64.82% White, 12.25% African American, 8.95% Asian, 1.84% Pacific Islander, 1.55% Native American, 3.55% from other races, 7.04% from two or more races. Hispanic or Latino of any race were 8.49% of the population. There were 23,792 households out of which 30.1% had children under the age of 18 living with them, 44.4% were married couples living together, 14.5% had a female householder with no husband present, 36.6% were non-families. 29.6% of all households were made up of individuals and 8.7% had someone living alone, 65 years of age or older.
The average household size was 2.38 and the average family size was 2.94. The population is spread out with 24.4% under the age of 18, 11.3% from 18 to 24, 29.4% from 25 to 44, 21.6% from 45 to 64, 13.2% who were 65 years of age or older. The median age was 35 years. For every 100 females, there were 95.8 males. For every 100 females age 18 and over, there were 93.9 males. The median income for a household in the city was $36,422, the median income for a family was $42,551. Males had a median income of $31,434 versus $26,653 for females; the per capita income for the city was $20,569. About 12.5% of families and 15.8% of the population were below the poverty line, including 23.5% of those under age 18 and 5.6% of those age 65 or over. The city was incorporated in 1996; the City of Lakewood contracted with the Pierce County Sheriff's Office for police services between 1996 and 2004. The city created the LPD in 2004. According to the City of Lakewood website, The Lakewood Police Department started independent police and patrol operations on November 1, 2004.
On November 29, 2009, four LPD officers were killed. Maurice Clemmons walked into the Parkland Forza Coffee shop at around 8:15 a.m. After approaching the counter, he started shooting. Dead at the scene were Sergeant Mark Renninger, 39, officers Tina Griswold, 40, Ronald Owens, 37, Greg Richards, 42; each of them had served with the department since its inception. Two baristas and several customers in the shop were not injured. Clemmons was killed by a Seattle police officer two days later; the shooting is believed to be the most deadly attack on law enforcement in the state of Washington, the second deadliest attack on law enforcement in the United States since the March 21, 2009 shootings that left four Oakland, California police officers dead. The four were the first Lakewood police officers to be killed in the line of duty since the department's establishment in 2004; the Clover Park School District operates all public schools within Lakewood. Lakewood is home to Pierce College Fort Steilacoom and Clover Park Tech
The 2001 Barangay Ginebra Kings season was the 23rd season of the franchise in the Philippine Basketball Association. Barangay Ginebra Kings makes a return trip to the PBA championship since 1997, the first time post-Sonny Jaworski era, playing against sister team San Miguel Beermen in the 2001 All-Filipino Cup finals; the Kings made it by winning twice over Purefoods Tender Juicy Hotdogs during the quarterfinal round with both victories by one-point margin. The Kings won their best-of-five semifinal series against three games to two. Ginebra center Jun Limpot played in his first finals appearance after eight years while Vergel Meneses is back in the PBA finals since his Sunkist days in 1995; the Barangay Ginebra Kings lost to San Miguel Beermen in six games. Mark Caguioa was named the season's Rookie of the Year. Team Manager: Ira Maniquis All-Filipino Cup Commissioner's Cup Governor's Cup
Ariel Rubinstein is an Israeli economist who works in Economic Theory, Game Theory and Bounded Rationality. Ariel Rubinstein is a professor of economics at the School of Economics at Tel Aviv University and the Department of Economics at New York University, he studied mathematics and economics at the Hebrew University of Jerusalem, 1972–1979. In 1982, he published "Perfect equilibrium in a bargaining model", an important contribution to the theory of bargaining; the model is known as a Rubinstein bargaining model. It describes two-person bargaining as an extensive game with perfect information in which the players alternate offers. A key assumption is; the main result gives conditions under which the game has a unique subgame perfect equilibrium and characterizes this equilibrium. Rubinstein was elected a member of the Israel Academy of Sciences and Humanities, a Foreign Honorary Member of the American Academy of Arts and Sciences in and the American Economic Association. In 1985 he was elected a fellow of the Econometric Society, served as its president in 2004.
In 2002, he was awarded an honorary doctorate by the Tilburg University. He has received the Bruno Prize, the Israel Prize for economics, the Nemmers Prize in Economics, the EMET Prize. and the Rothschild Prize. Bargaining and Markets, with Martin J. Osborne, Academic Press 1990 A Course in Game Theory, with Martin J. Osborne, MIT Press, 1994. Modeling Bounded Rationality, MIT Press, 1998. Economics and Language, Cambridge University Press, 2000. Lecture Notes in Microeconomic Theory: The Economic Agent, Princeton University Press, 2006. Economic Fables, Open Book Publishers, 2012. AGADOT HAKALKALA, Zmora, Bitan, 2009. List of Israel Prize recipients Personal Web site Nash lecture Russ. "Rubinstein on Game Theory and Behavioral Economics". EconTalk. Library of Economics and Liberty