Create the program to return the orthogonal basis and orthonormal basis

Question

Nguyet Nguyen 2022-8-13

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编辑： Torsten 2022-8-14

Write a program to input any number of vectors in R^n and return return the orthogonal basis and orthonormal basis of the subspace spanned by these vectors using Gram Schmidt process. Can someone check if I do right

for j= 1:n

v = A (: , j) ;

for i = 1: j-1

R(i,j) = Q (:, i )' * A ( :, j) ;

v = v - R ( i ,j ) * Q ( :, i);

end

R ( j, j ) = norm ( v );

Q (:, j) = v/R ( j, j ) ;

end

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Answer 1

Torsten 2022-8-13

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https://ww2.mathworks.cn/matlabcentral/answers/1778830-create-the-program-to-return-the-orthogonal-basis-and-orthonormal-basis#answer_1026025

编辑：Torsten 2022-8-13

在 MATLAB Online 中打开

As far as I can see, you only return the orthonormal basis from the Gram Schmidt process in the matrix Q.

And saving the norms is superfluous if you have to return the orthonormal basis, too.

n = 2;
A = [1 3; 4 -7; -1 -12];
Q = zeros(size(A));
QN = zeros(size(A));
for j= 1:n 
    v = A (:,j) ; 
    for i = 1: j-1
        rij = Q(:,i).' * A(:,j) / (Q(:,i).' * Q(:,i));
        v = v - rij * Q(:,i);
    end
    Q(:,j) = v;
    QN(:,j) = v/norm(v);
end
sqrt(Q.'*Q)
ans = 2×2
    4.2426         0
         0   13.8784
QN.'*QN
ans = 2×2
    1.0000         0
         0    1.0000

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Nguyet Nguyen 2022-8-14

tks u so much

Torsten 2022-8-14

编辑：Torsten 2022-8-14

If the resulting vector v is the zero vector, you shouldn't include it as column in Q. This will happen if vi is already in the span of v1,...,v_i-1. And this will definitely happen if more than n vectors are supplied - as is possible according to the formulation of the assignment.

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