Unitary transformation and Orthogonal transformation. Difference between unitary and orthogonal transformation is whether it is in Complex or Real 
Euclidean Space. 
So, If the matrix is orthogonal, does that mean that transformation of this matrix is 
orthogonal?
Is there any orthogonal matrix having complex numbers as entries?  
 A: A matrix is defined to be orthogonal if it preserves a symmetric, nondegenerate bilinear form. These can be defined over any field. Over the real numbers, an additional restriction is usually imposed making the form also positive definite (like the dot product, means $(x,x)>0$ whenever $x\neq 0$), but bilinear forms can only be positive definite over a field where such a thing as positivity exists. In the usual sense, orthogonal matrices are the ones that preserve the dot product. Orthogonal matrices also exist over the complex numbers that preserve the dot product. However, the dot product itself is not so interesting over the complex numbers, at least not to analysts. The orthogonal matrices themselves are important in Lie group theory.
Unitary matrices don't preserve dot product, they preserve a Hermitian form, which is not bilinear but is instead conjugate linear in the second argument, and it is positive definite in the same sense as the dot product.
A: To your first question: yes, the transformation on $\Bbb R$ associated with an orthogonal matrix is orthogonal.
To your second question: generally, matrices with complex (i.e. non-real) numbers are only used to describe transformations on vector spaces over $\Bbb C$.  So no: matrices with complex (non-real) entries do not describe orthogonal transformations (which refers specifically to a type of transformation on real vector spaces).
