svd(T) = u sigma v Here I understand meaning of each and every term and why SVD is important.
But I am failing to interpret this equation from Linear Algebra glasses.
When I have learnt about Linear Algebra, there was one thing common in all sources that is to view matrix is as Basis Vectors (or transformation matrix)
T v = λ u T = transformation matrix v = some vector to be transformed after applying T u = transformed unit vector λ = scale of transformation
But I am failing to relate this when it comes to SVD.
T = u sigma v
(Note: I am not looking for a answer what svd means and what each term means. I am looking for an answer precisely on following confusion)
To be more precise:
Represent our data as Transformation Matrix:
Our data = m*n matrix = T
Apply transformation matrix T (that is our data) on some vector:
T * some-vector = new-rotated-unit-vector * scaling-factor
we get the same effect as above, by calling 3 different transformations (rotation-scaling-rotation) denoted by:
T = U.sigma. V ....... svd
So it means, on any vector v, we can apply T (our data matrix) or 3 transformations (U.sigma.V) and we can have the same effect.
So far so good when we see above operations only from transformation perspective.
Now suddenly, we change whole perspective. It is no more a transformation perspective.
As per new perspective, sigma-matrix also have other meaning apart from scaling-matrix which suggests which axis has highest variation to project our data.
This is so confusing. We used our data as transformation matrix T and decomposed it on 3 different matrices. This is okay no issues.
Now we are saying, on one of these decomposed matrix, we can project our data (our data which we used as transformation matrix) onto that.
I am unable to match these 2 perspective and that is my problem.