Unitary matrix

A square matrix $ A = \| a _ {ik} \| _ {1} ^ {n} $ over the field $ \mathbf C $ of complex numbers, whose rows form an orthonormal system, i.e.

$$ a _ {i1} \overline{a}\; _ {k1} + \dots + a _ {in} \overline{a}\; _ {kn} = \ \left \{

$ i, k = 1 \dots n $. In a unitary space, transformation from one orthonormal basis to another is accomplished by a unitary matrix. The matrix of a unitary transformation relative to an orthonormal basis is also (called) a unitary matrix. A square matrix $ A $ with complex entries is unitary if and only if it satisfies any of the following conditions:

1) $ A ^ {*} A = E $;

2) $ AA ^ {*} = E $;

3) $ A ^ {*} = A ^ {-} 1 $;

4) the columns of $ A $ form an orthonormal system (here $ A ^ {*} $ is the conjugate transposed of $ A $).

The determinant of a unitary matrix is a complex number of modulus one.

References

[a1]	W. Noll, "Finite dimensional spaces" , M. Nijhoff (1987) pp. 63
[a2]	W.H. Greub, "Linear algebra" , Springer (1975) pp. 329