Unitary group

In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields.

In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group.

The unitary group U(n) is a real Lie group of dimension n². The Lie algebra of U(n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator.

The general unitary group (also called the group of unitary similitudes) consists of all matrices A such that A^∗A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix.

Properties

Since the determinant of a unitary matrix is a complex number with norm , the determinant gives a group homomorphism

U2 (disambiguation)

U2 is an Irish rock band.

U2 or U-2 may also refer to:

Military

Railway

Other uses

SM U-2 (Germany)

SM U-2 was a German U-boat built for the Imperial German Navy. Only one of the type, sometimes called German Type U 2 submarine, was built. U-2 was ordered from Kaiserliche Werft of Danzig on 4 March 1906, launched on 18 June 1908, and commissioned into the Imperial German Navy on 18 July 1908. She conducted no war patrols and spent World War I as a training platform.

After Germany's surrender, she was decommissioned on 19 February 1919 and sold for shipbreaking to Stinnes on 3 February 1920.

Citations

References

Spy aircraft

Spy aircraft may refer to:

Lockheed U-2

The Lockheed U-2, nicknamed "Dragon Lady", is a single-jet engine, ultra-high altitude reconnaissance aircraft operated by the United States Air Force (USAF) and previously flown by the Central Intelligence Agency (CIA). It provides day and night, very high-altitude (70,000 feet; 21,000 m), all-weather intelligence gathering. The U-2 has also been used for electronic sensor research, satellite calibration, and communications purposes.

Early versions of the U-2 were involved in several events through the Cold War, being flown over the Soviet Union, China, Vietnam, and Cuba. In 1960, Gary Powers was shot down in a CIA U-2A over the Soviet Union by a surface-to-air missile. Another U-2, piloted by Major Rudolf Anderson, Jr., was lost in a similar fashion in the Cuban Missile Crisis of 1962.

The U-2 is one of a handful of aircraft types to have served the USAF for over 50 years. The newest models (TR-1, U-2R, U-2S) entered service in the 1980s. The current model, the U-2S, received its most recent technical upgrade in 2012. They have taken part in post-Cold War conflicts in Afghanistan and Iraq, and supported several multinational NATO operations.