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  • U = UU^--- = I_n\,


  • \, 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^--- \,.


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 angle
so also a unitary matrix ''U'' satisfies
:\langle Ux, Uy angle = \langle x, y angle
for all ''complex'' vectors ''x'' and ''y'', where <.,.> stands now for the standard Inner Product on C''n''. If A \, is an ''n'' by ''n'' matrix then the following are all equivalent conditions:

#A \, is unitary
  • \, is unitary

    • # the columns of A \, form an Orthonormal Basis of C''n'' with respect to this inner product

# the rows of A \, form an orthonormal basis of C''n'' with respect to this inner product
# A \, is an Isometry with respect to the norm from this inner product

It follows from the isometry property that all Eigenvalue s 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



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 .

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


SEE ALSO