Hi,
Suppose you are applying SVD (Singular Value Decomposition) to a 2D image and rewrite the image as A=u*s*transpose(v). Note that S is diagonal. In some references (https://en.wikipedia.org/wiki/Singular_value_decomposition)it is argued that the matrix s can be interpreted as a scaling matrix. Can you assign non-zero values to some off-diagonal elements of s? Does it make sense to assign non-zero values to off-diagonal elements? Does it not violate the inherent independence of basis vectors (for example i, j, k in Cartesian 3D space)?
Your cooperation is highly appreciated.
I do not see how it would make sense to make off-diagonal elements of S non-zero. You would be changing the image A to no identifiable purpose.
In particular, you would be replacing the diagonal matrix S with (S + S'), where S' has zero on its diagonal. The new image would be
A' = U (S+S') transpose(V) = A + U S' transpose(V)
Is it possible that the addition of (U S' transpose(V)) to the original matrix A is of some use? I see no reason to suppose that it would be of general use.
Ron
Dear Amir
In the formal definition of SVD the matrix S must be diagonal. For more explanation about SVD you can see section 7.3 in the book "Matrix Analysis" by Horn and Johnson.
Regards
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