2014/04/22 | Improve Society

Orthogonal and unitary matrices

A real matrix $A$ is called an orthogonal matrix if its transpose is the same as its inverse, i.e. if $A^T = A^{-1}$ or $A^TA= I$ .

A complex matrix $A$ is called a unitary matrix if its complex conjugate transpose is the same as its inverse, i.e. if $\bar{A}^T = A^{-1}$ or $\bar{A}^TA = I$ . It should be noted that a real unitary matrix is an orthogonal matrix.

直交行列とユニタリ行列

　実行列 $A$ はその転置行列が自身の逆行列と同じ場合，すなわち仮に $A^T = A^{-1}$ または $A^TA= I$ ならば 直交行列 と呼びます．

　複素行列 $A$ は自身の複素共軛転置行列が逆行列と同じなら，すなわち仮に $\bar{A}^T = A^{-1}$ または $\bar{A}^TA = I$ ならば ユニタリ行列 と呼びます．実ユニタリ行列は直交行列であることに注意が必要です．