- Unitary matrix
In
mathematics , a unitary matrix is an "n" by "n" complex matrix "U" satisfying the condition:U^* U = UU^* = I_n, where I_n, is the
identity matrix and U^* , is theconjugate transpose (also called theHermitian adjoint ) of "U". Note this condition says that a matrix "U" is unitary if and only if it has an inverse which is equal to itsconjugate transpose U^* ,:U^{-1} = U^* ,;
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,:langle Gx, Gy angle = langle x, y angleso also a unitary matrix "U" satisfies:langle Ux, Uy angle = langle x, y anglefor all "complex" vectors "x" and "y", where langlecdot,cdot angle stands now for the standardinner product on C"n". If U , is an "n" by "n" matrix then the following are all equivalent conditions:#U , is unitary
#U^* , is unitary
# the columns of U , form anorthonormal basis of C"n" with respect to this inner product
# the rows of U , form an orthonormal basis of C"n" with respect to this inner product
# U , is anisometry with respect to the norm from this inner productIt follows from the isometry property that all
eigenvalue s of a unitary matrix are complex numbers ofabsolute value 1 (i.e., they lie on theunit circle centered at 0 in thecomplex plane ). The same is true for thedeterminant .All unitary matrices are normal, and the
spectral theorem therefore applies to them. Thus every unitary matrix "U" has a decomposition of the form:U = VSigma V^*;
where "V" is unitary, and Sigma is diagonal and unitary.
For any "n", the set of all "n" by "n" unitary matrices with matrix multiplication form a group.
Properties of unitary matrices
* U is invertible
* U^{-1}=U^*
* |det(U)| = 1
* U^* is unitary
* Unitary matrices preserve length Ux|_2=|x|_2ee also
*
Hermitian matrix
*symplectic matrix
*unitary group
*special unitary group
*unitary operator
*defect of a unitary matrix
*matrix decomposition External links
* [http://mathworld.wolfram.com/UnitaryMatrix.html Unitary Matrix from Mathworld ]
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