Unitary
Adjective and noun used in mathematics, physics, and political science: describes operators preserving inner products, groups of such matrices, and centralized state organization.
Unitary is an adjective and noun with important meanings in mathematics, physics, computer science and political science. Broadly, it denotes operations or structures that preserve a fundamental form of 'unity' — in mathematics and physics that means preserving inner products and lengths; in government it refers to a single central authority.
Mathematics and linear algebra
In linear algebra a unitary operator on a complex inner-product space is a linear map U that preserves the inner product: for all vectors x and y, <Ux,Uy> = <x,y>. For matrices this property is equivalent to U*U = I, where U* is the conjugate transpose (adjoint) and I is the identity. A unitary matrix is necessarily invertible and its eigenvalues lie on the complex unit circle (have modulus 1).
Groups and notable families
- The unitary group U(n) is the set of all n×n unitary matrices under multiplication; it is a compact Lie group.
- The special unitary group SU(n) consists of unitary matrices with determinant 1 and plays a central role in geometry and theoretical physics.
Physics and quantum mechanics
Unitary operators govern reversible, probability-preserving evolution in quantum mechanics. Time evolution by the Schrödinger equation is represented by a unitary operator on the system's Hilbert space. In quantum computing, logical gates are implemented as unitary matrices acting on qubit state vectors.
Other uses and contrasts
The term also appears outside mathematics: a unitary state in political science denotes a system where central authorities hold primary power and subnational units exercise delegated powers. Mathematically, unitary transformations are distinct from orthogonal ones only in that orthogonal matrices are the real-valued special case of unitary matrices; unitary is not the same as Hermitian, which denotes self-adjoint operators with real eigenvalues.
Key practical applications include Fourier and other integral transforms (which are unitary up to normalization), quantum algorithms, signal processing, and the mathematical classification of symmetries in physics.
Questions and answers
Q: What is the definition of unitary?
A: Unitary refers to a type of government or state where power is centralized and the central government has sole authority and control over all other aspects of governance.
Q: What is an example of a unitary state?
A: The United Kingdom is an example of a unitary state because it has a centralized government in Westminster which has authority and control over all aspects of the country.
Q: What is the opposite of a unitary state?
A: The opposite of a unitary state is a federal state where power is divided between the central government and the regional or state governments.
Q: How does a unitary state differ from a confederation?
A: A unitary state differs from a confederation in that in a confederation, power is decentralized and is primarily held by the regional or state governments rather than by a central government.
Q: What are the advantages of a unitary system of government?
A: The advantages of a unitary system of government include a more efficient and streamlined decision-making process, greater accountability, and a stronger sense of national unity.
Q: What are some of the potential disadvantages of a unitary system of government?
A: Potential disadvantages of a unitary system of government include a lack of local autonomy and representation, potentially oppressive or unresponsive central government, and a lack of protection for minority groups or regions.
Q: Are there any countries where there is a mix of unitary and federal systems of government?
A: Yes, some countries, such as China and Spain, have a mix of unitary and federal systems of government where power is decentralized to some degree but ultimate authority and control rest with the central government.
Related articles
Author
AlegsaOnline.com Unitary Leandro Alegsa
URL: https://en.alegsaonline.com/art/102939