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In mathematics, a complex square matrix ''U'' is unitary if its conjugate transpose ''U''∗ is also its inverse – that is, if :: where ''I'' is the identity matrix. In physics, especially in quantum mechanics, the Hermitian conjugate of a matrix is denoted by a dagger (†) and the equation above becomes :: The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes. ==Properties== For any unitary matrix ''U'' of finite size, the following hold: *Given two complex vectors ''x'' and ''y'', multiplication by ''U'' preserves their inner product; that is, :. *''U'' is normal *''U'' is diagonalizable; that is, ''U'' is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus ''U'' has a decomposition of the form :: :where ''V'' is unitary and ''D'' is diagonal and unitary. * . * Its eigenspaces are orthogonal. * ''U'' can be written as , where ''e'' indicates matrix exponential, is the imaginary unit and ''H'' is an Hermitian matrix. For any nonnegative integer ''n'', the set of all ''n''-by-''n'' unitary matrices with matrix multiplication forms a group, called the unitary group U(''n''). Any square matrix with unit Euclidean norm is the average of two unitary matrices. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「unitary matrix」の詳細全文を読む スポンサード リンク
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