Orthogonality of a 4x4 DCT matrix

Question

Anonym 2020-12-7

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评论： Bjorn Gustavsson 2022-9-26

I am working on a MATLAB task which deals with stain removal and Discrete Cosine tranformation.

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:

5000   0.5000   0.5000   0.5000   
6533   0.2706  -0.2706  -0.6533
5000  -0.5000  -0.5000   0.5000  
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.

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Answer 1

Bjorn Gustavsson 2020-12-8

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What does it mean that a matrix is orthogonal?

What is the condition for two vectors to be orthogonal?

Answer these two questions and the easiest method will become obvious to you.

HTH

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Answer 2

Farooq 2022-9-24

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在 MATLAB Online 中打开

Orthogonality of a matrix means that the matrix multiplied by its inverse is equal to the identity matrix.

matrix * matrix ' = I

In MATLAB you can code this for example for a matrix "x"

if x*x' == eye(size(x))
    y = true
else
    y = false
end

I hope this helps.

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Bjorn Gustavsson 2022-9-26

Well, your code is OK but it doesn't correspond to your phrasing, and your phrasing is a bit "too generous" - every matrix multiplied by its inverse should result in the identity-matrix, surely?

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