Orthogonality of a 4x4 matrix

Question

I am working on a MATLAB task which deals with stain removal.

What am I doing? I have been given a 4x4 matrix. I have then been told that it may well be orthogonal. I have to make it prove that the DCT matrix is actually orthogonal.

This is the given DCT matrix:

0.5000   0.5000   0.5000   0.5000   
0.6533   0.2706  -0.2706  -0.6533
0.5000  -0.5000  -0.5000   0.5000  
0.2706  -0.6533   0.6533  -0.2706

Here's the code:

function [U, C, G] = UFGDCT(N)
%
%   Compute the matrices for DCT (-?-)
%
%   U is the unitary "in-between" matrix
%   C is the matrix of the DCT
%   G is the inverse of F
%
    C = zeros(N);
    for row = 0:N-1
        for col = 0:N-1
            C(row+1, col+1) = cos(pi*row*(col+(1/2))/N);
        end
    end
    for cols = 0:N-1
        C(1,cols+1) = C(1,cols+1)/sqrt(2);
    end
    C = C*sqrt(2/N);
    U = C;
    G = C';
end

How can I do it in the simplest way? I have tried to search about finding orthogonality of a matrix, but didn't get the luch. I could not find anything that could be helpful.

RESULT:

mat*mat'
mat'*mat

ans =

    1.0000         0         0         0
         0    1.0001         0         0
         0         0    1.0000         0
         0         0         0    1.0001


ans =

    1.0000    0.0000   -0.0000   -0.0000
    0.0000    1.0000   -0.0000   -0.0000
   -0.0000   -0.0000    1.0000    0.0000
   -0.0000   -0.0000    0.0000    1.0000

Answer 1

If you just want to check the orthogonality, you can use mat*mat' and mat'*mat to see if it is an identity matrix

>> mat*mat'
ans =

   1.0000        0        0        0
        0   1.0001        0  -0.0000
        0        0   1.0000  -0.0000
        0  -0.0000  -0.0000   1.0001

>> mat'*mat
ans =

   1.0000e+00  -2.4018e-17  -3.7377e-18  -2.5250e-05
  -2.4018e-17   1.0000e+00  -2.5250e-05  -3.7377e-18
  -3.7377e-18  -2.5250e-05   1.0000e+00  -2.4018e-17
  -2.5250e-05  -3.7377e-18  -2.4018e-17   1.0000e+00

You will see that a tolerance should be set to determine if mat is orthogonal

>> mat*mat'-eye(4)
ans =

            0            0            0            0
            0   5.0500e-05            0  -3.7377e-18
            0            0            0  -2.7756e-17
            0  -3.7377e-18  -2.7756e-17   5.0500e-05

>> mat'*mat-eye(4)
ans =

   2.5250e-05  -2.4018e-17  -3.7377e-18  -2.5250e-05
  -2.4018e-17   2.5250e-05  -2.5250e-05  -3.7377e-18
  -3.7377e-18  -2.5250e-05   2.5250e-05  -2.4018e-17
  -2.5250e-05  -3.7377e-18  -2.4018e-17   2.5250e-05

Data

mat = [0.5000   0.5000   0.5000   0.5000   
0.6533   0.2706  -0.2706  -0.6533
0.5000  -0.5000  -0.5000   0.5000  
0.2706  -0.6533   0.6533  -0.2706];