Difference between Singular Value Decomposition and Smith Decomposition
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Siddhanth Sunil Shah
le 8 Jan 2022
Commenté : Christine Tobler
le 12 Jan 2022
Is there any difference between Singular Value Decomposition and Smith Decomposition?
Is there any relation between the two or they are one and the same thing?
I know Smith decomposition can be applied to only square matrices (in Matlab) and SVD is applicable to any matrix.
However if we take any mxn matrix and derive the SVD and Smith forms, then would they be the same or different?
6 commentaires
Christine Tobler
le 12 Jan 2022
Thank you both for the explanations, I have passed this information on to colleagues in the Symbolic Toolbox.
Réponse acceptée
John D'Errico
le 8 Jan 2022
Modifié(e) : John D'Errico
le 8 Jan 2022
How does this apply to MATLAB? This really is a question purely about linear algebra.
Are they the same? No, they are not. Yes, they look somewhat alike in what they return, ALMOST.
The smithForm applies ONLY to square matrices. But also, it applies ONLY to integer valued matrices. READ THE HELP!
help smithForm
Next, try it out. Don't know if they are different? TRY IT.
A = magic(3)
[U,S,V] = svd(A)
[U2,S2,V2] = smithForm(A)
They don't look the same to me. But then I do need a new set of glasses.
Finally, if you try using smithform even on a square matrix that has non-integer elements, it will fail.
A = randn(2)
svd(A)
smithForm(A)
The Smith normal form is designed to solve a totally different set of problems compared to when one would use the svd. They are different tools, with different properties, different purposes, and different results.
2 commentaires
Paul
le 8 Jan 2022
Modifié(e) : Paul
le 8 Jan 2022
As stated in the Help, the function smithForm() also applies to symbolic, polynmomial matrices, not just integer valued matrices.
While true that the smithForm() function is limited to square matrices, it's worth pointing out that that is a limitation of Matlab's implementation and not of the Smith form in general, as shown in the wikipedia link.
As shown in the Help, this line should be
%[U2,S2,V2] = smithForm(A)
[U2,V2,S2] = smithForm(A)
which, of course, wouldn't change the conclusion that Smith and SVD are different.
As an aside, I'm not too surprised that smithForm() accepts a double as input, but I am surprised it returns double (not sym) as output.
John D'Errico
le 8 Jan 2022
Agreed, as part of the symbolic TB, it would make sense for it to return a symbolic result.
I do wish they had chosen different names for the return variables. While it is completely irrelevant what they call the variables, people will see U,S,V, and get confused, thinking the two are the same.
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