svd

SVD is a matrix factorization technique.

For a given matrix, there exists a decomposition of it which can be written as follows: $M = U . \Sigma . V^*$

• M = the input matrix of dimension m x n.
• U = m x m unitary matrix
• $\Sigma$ = m x n diagonal matrix
• V = n x n unitary matrix
• V* = conjugate transpose of V

Basic idea behind SVD is that the transformation operation performed by a matrix M can be viewed as a sequence of the following transformations:

1. V* represents the rotation operation applied onto to a unit hypersphere.
2. $\Sigma$ then represents the scaling across the co-ordinate axes.
3. U represents another rotation after scaling operation.

SVD is most often used in finding eigen values and vectors of the matrix. For example: columns of V are eigenvectors of $M^*.M$. columns of U are eigenvectors of $M.M^*$ and non-zero elements of $\Sigma$ are sqrt of their eigen values.