【Unitary Matrix】 么正矩陣、酉矩陣、單一正規化矩陣

From Wikipedia, the free encyclopedia

Jump to: navigation, search

In mathematics, a unitary matrix is a (square) complex matrix satisfying the condition

where is the identity matrix in n dimensions and is the conjugate transpose (also called the Hermitian adjoint) of . This condition implies that a matrix is unitary if and only if it has an inverse which is equal to its conjugate transpose

A unitary matrix in which all entries are real is an orthogonal matrix. Just as an orthogonal matrix preserves the (real) inner product of two real vectors,

so also a unitary matrix satisfies

for all complex vectors x and y, where is the standard inner product on .

If is an matrix then the following are all equivalent conditions:

1. is unitary
2. is unitary
3. the columns of form an orthonormal basis of with respect to this inner product
4. the rows of form an orthonormal basis of with respect to this inner product
5. is an isometry with respect to the norm from this inner product
6. is a normal matrix with eigenvalues lying on the unit circle.