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 the conjugate transpose (also called the Hermitian 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 its conjugate 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 \rangle = \langle x, y \rangle

so also a unitary matrix U satisfies

\langle Ux, Uy \rangle = \langle x, y \rangle

for all complex vectors x and y, where \langle\cdot,\cdot\rangle stands now for the standard inner product on Cn. If U \, is an n by n matrix then the following are all equivalent conditions:

  1. U \, is unitary
  2. U^* \, is unitary
  3. the columns of U \, form an orthonormal basis of Cn with respect to this inner product
  4. the rows of U \, form an orthonormal basis of Cn with respect to this inner product
  5. U \, 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.

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 = V\Sigma V^*\;

where V is unitary, and Σ is diagonal and unitary.

For any n, the set of all n by n unitary matrices with matrix multiplication form a group.


Contents

Properties of unitary matrices

See also

External links

References

  1. ^ R. Shankar, Principles of Quantum Mechanics, 2nd Ed., pg. 39.