Talk:Orthogonal Procrustes problem
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Why it must be and not edit
Minimizing the Frobenius norm of is equivalent to maximizing the trace of , since . Now, using the SVD , we get . Now, is orthogonal and is diagonal and positive, and thus the trace get's maximized if is the identity, therefore . No assumption on the and concerning orthogonality has been made, only the orthogonality of and has been used. (see also Golub/van Loan or the Schönemann paper. Ezander (talk) 11:34, 24 March 2010 (UTC)
A proof candidate edit
Let be an orthogonal matrix and be an arbitrary matrix. Since
minimizing the Frobenius norm of the difference is equivalent to maximizing . Let be the singular value decomposition of , where and are orthogonal and is diagonal. Now
where is an arbitrary orthogonal matrix. Furthermore,
since is diagonal. Now, , since the columns of are unit vectors. Over arbitrary matrices , without the orthogonality constraint, but with the diagonal constraint, maximizes , since the diagonal elements of are non-negative. However, the identity matrix is an orthogonal matrix with this property. Thus , from which it follows
.
Errors in page edit
A and B have the same size. Let's say m x n. Then omega and R must have size m x m, since we are multiplying A on the right by omega.
The "Solution" section then says to take the SVD of A^T * B. That has size n x n. So U, sigma, and V all have size n x n. So V * U^T has size n x n.
Clearly there are errors in here somewhere. I think if the objective was to minimize the norm of A*omega - B it would be correct, because then n x n would be the correct size for omega.
I don't know the markup used by Wikipedia. Could someone fix this? — Preceding unsigned comment added by 130.33.205.56 (talk) 00:06, 17 November 2015 (UTC)
- I agree, there's this inconsistency in the page. I was actually trying to fix it but I realised then that the whole proof needs changing so I'll do that when I have more time. --Roberto→@me 13:29, 17 May 2016 (UTC)
- I see this issue has not yet been addressed. The size of A, B, Ω, R should be specified in the article. The simplest way is to make them all square so that none of the issues mentioned here arise. I guess ideally we would have the general problem, solution, and proof. Not sure if we have the time for that, so please let me know if I can edit it at least to be correct for square matrices A, B. Sunbeam44 (talk) 20:28, 15 March 2024 (UTC)