The author provides a factorization of unitary \(n\times n\) matrices. The unitary group \(U(n)\) is the group of automorphisms of the Hilbert space \((\mathbb{C}^n,\langle.|.\rangle)\) where \(\langle.|.\rangle\) is the Hermitian scalar product. For \(A_n\in U(n)\), \(A^*_n\) denotes the adjoint matrix. Then \(A^*_n A_n= I_n\), where \(I_n\) is the \(n\times n\) unit matrix. Thus \(\text{det\,} A_n= e^{i\varphi}\), where \(\varphi\) is a phase, and \(\dim_{\mathbb{R}}U(n)= n^2\). The following is proved:
1. Any unitary matrix \(A_n\in U(n)\) can be factored into an ordered product of \(2n- 1\) matrices of the form \(A_n = d_n O_n d^1_{n-1} O^1_{n-1}\cdots d^{n-2}_2 O^{n-1}_2 d^{n-1}_1\), where \(d^k_{n-k}\) are diagonal phase matrices and \(O^k_{n-k}\) are orthogonal matrices whose columns are generated by real \((n-k)\)-dimensional unit vectors.
2. If the condition \(\sum^{n(n+1)/2}_{i=1}\varphi_i= 0\) is imposed on the arbitrary phases \(\varphi_i\) entering the parametrization of \(A_n\) the factorization of \(\text{SU}(n)\) matrices is obtained.
3. If \(w_n= O_n d^1_{n-1} O^1_{n-1}\cdots d^{n-2}_2 O^{n-1}_2 d^{n-1}_1= O_n d^1_{n-1} w_{n-1}\), then \(W_n= w^*_n d_n w_n\) is one of the possible Weyl representations of unitary matrices.
4. If all the phases \(\varphi_i\) entering \(A_n\) are zero or \(\pi\) \((i= 1,\dots, n(n+ 1)/2)\) one gets the factorization of the rotation group \(O(n)\). The factorization of the special group \(\text{SO}(n)\) is obtained when an even number of phases has the value \(\pi\).
These results can be used to parametrize complex Hadamard matrices and to find the Laplace-Beltrami operators on unitary groups.