There are several ways to generate a low-rank approximant for each capable block of the original matrix. A class of analytical methods are including but not limited to Taylor series approximation, multipole expansion, and interpolation. Also, algebraic techniques are singular value decomposition (SVD), pseudo-skeletal approximation, cross approximation and its variants, rank-revealing LU and QR factorization.
Dear Rezaeipour
Can you, perhaps, indicate what your question is? Perhaps if we (the folks in the community) knew your intended purpose for the low rank approximation, we could give you guidance on selecting such a matrix. For many purposes, low rank approximation via the SVD is the most useful approach, but perhaps your application would favor another approach.
Jeff
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