Generate an orthonormal matrix H mHxNh rank(H)=r

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Domenico de Falco 2024-2-10

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https://ww2.mathworks.cn/matlabcentral/answers/2080711-generate-an-orthonormal-matrix-h-mhxnh-rank-h-r

编辑： John D'Errico 2024-2-12

% Problem : Find an orthonormal matrix with dimension mH x nH and r=rank(H)

% We define orthonormal a matrix A(mH x nH) : A'*A=InH
clear; close all; clc
mH=5; nH=3; r=2;

% Case 1 mH>NH

A = rand(mH,nH);
[W,L,V]=svd(A,'econ')
ll=diag(L);
ll=ll(1:r);
Lnew=diag(ll);
Wnew=W(:,1:r)
Vnew=V(:,1:r)
Anew=Wnew*Lnew*Vnew';
H=Vnew
size(H)
V'*V % check orthonormality

% Caso 2 m<b: how to do ?

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Domenico de Falco 2024-2-11

编辑：Domenico de Falco 2024-2-11

pardon Case 2: mH<nH: how to do ?

John D'Errico 2024-2-12

Surely you read my answer. I think you do not understand my answer though. I'll explain what happens when mH<nH in that answer.

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Answer 1

John D'Errico 2024-2-10

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2080711-generate-an-orthonormal-matrix-h-mhxnh-rank-h-r#answer_1406311

编辑：John D'Errico 2024-2-12

在 MATLAB Online 中打开

Seems trivial enough. Not possible, at least not for this case. But trivially not possible.

mH=5; nH=3; r=2;

So a 5x3 array, of rank 2. That part is trivial. HOWEVER, what is NOT possible is for the product A'*A to result in an identity matrix. NOT possible, EVER. For example...

A1 = randn(5,2); A2 = randn(2,3);
A = A1*A2;

A has rank 2, because it is the product of two rank 2 random matrices.

rank(A)
ans = 2

Now trivially, we can orthogonalize the columns of A.

Aorth = orth(A)
Aorth = 5×2
   -0.8498   -0.3246
    0.2709    0.2936
   -0.4471    0.8382
   -0.0130    0.0917
    0.0661    0.3123

But, since A has rank only 2, there are only two basis vectors we can ever get out of it. They are orthogonal. But the product Aorth'*Aorth will be only a 2x2 identity, not a 3x3 identity.

Aorth'*Aorth
ans = 2×2
    1.0000   -0.0000
   -0.0000    1.0000

Think about it. If the matrix A is of rank 2, then A'*A can never have rank more than 2. Sorry, but it is flat out impossible for the product to be a 3x3 identity matrix, which has rank THREE. Basic linear algebra here.

Again, finding a 5x3 matrix of rank 2, this is easy enough. Then you can orthogonalize the columns trivially (just use orth). But since it has rank only 2, there will be only 2 columns in the result. As much as you want to do the mathematically impossible, the laws of linear algebra will get in your way.

Why was your code completely bogus? Simple enough.At the end, you do this:

% V'*V % check orthonormality

And of course it has the desired property, BUT, you did not use all the columns of V to build Anew. That is the flaw in your code, and why it is useless.

In the case of m<b, we have no clue, since you don't tell us what m and b are, and how they impact this.

Edit: Apparently the question was about mH<nH. There is still a fundamental flaw in your goal.

Can you generate an array of size mH by nH, of rank r? Again, this is trivial to achieve. For example...

mH=3; nH=5; r=2;

So the array will have 3 rows, 5 columns. But rank will still be r. Again, the scheme is trivial. Just use a product of two matrices as I did before.

A1 = randn(mH,r); A2 = randn(r,nH);
A = A1*A2
A = 3×5
   -1.2321    0.4063   -1.2326    2.9057    0.7508
    1.1452    0.2425    2.1002   -3.8171   -0.7325
   -0.3892    0.6921    0.4784   -0.0970    0.2056
size(A)
ans = 1×2
     3     5
rank(A)
ans = 2

So A is a 3x5 matrix, of rank 2. While we can orthogonalize the columns of A, the result will have only TWO independent, orthogonal basis vectors. You CANNOT, EVER have more than 2 such basis vectors.

Aorth = orth(A)
Aorth = 3×2
    0.6041    0.4588
   -0.7966    0.3242
   -0.0229    0.8273

Now, we can show the columns of Aorth are indeed orthonormal.

Aorth'*Aorth
ans = 2×2
    1.0000   -0.0000
   -0.0000    1.0000

But again, it is trivially a flaw in your request, that Aorth CANNOT be mH by nH, so it cannot ever be 3x5. Again, just basic linear algebra. It is not a question of how I generated the matrices. There is no other scheme to generate the matrices.