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Encyclopedia > Unitary matrix

In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... For the square matrix section, see square matrix. ...

where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse which is equal to its conjugate transpose U*. In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

A unitary matrix in which all entries are real is the same thing as an orthogonal matrix. Just as an orthogonal matrix G preserves the (real) inner product of two real vectors, In linear algebra, an orthogonal matrix is a square matrix G whose transpose is its inverse, i. ... In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

so also a unitary matrix U satisfies

for all complex vectors x and y, where <.,.> stands now for the standard inner product on Cn. If A is an n by n matrix then the following are all equivalent conditions: In mathematics, an inner product space is a vector space with additional structure, an inner product, scalar product or dot product, which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

1. A is unitary
2. A* is unitary
3. the columns of A form an orthonormal basis of Cn with respect to this inner product
4. the rows of A form an orthonormal basis of Cn with respect to this inner product
5. A is an isometry with respect to the norm from this inner product

It follows from the isometry property that all eigenvalues of a unitary matrix are complex numbers of absolute value 1 (i.e. they lie on the unit circle centered at 0 in the complex plane). The same is true for the determinant. In mathematics, an orthonormal basis of an inner product space V(i. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... The graph of the absolute value function In mathematics, the absolute value (or modulus) of a real number is its numerical value without regard to its sign. ... Illustration of a unit circle. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

All unitary matrices are normal, and the spectral theorem therefore applies to them. A complex square matrix A is a normal matrix iff where A* is the conjugate transpose of A (if A is a real matrix, this is the same as the transpose of A). ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...

A unitary matrix is called special if its determinant is 1.

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