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
How to Cite This Entry:
Unitary matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unitary_matrix&oldid=49083
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article