Question

I am doing a project which requires me to normalize a sparse NxNmatrix. I read somewhere that we can normalize a matrix so that its eigen values lie between [-1,1] by multiplying it with a diagonal matrix D such that N = D^{-1/2}*A*D^{-1/2}.

But I am not sure what D is here. Also, is there a function in Matlab that can do this normalization for sparse matrices?

Answer 1

It’s possible that I am misunderstanding your question, but as it reads it makes no sense to me.

A matrix is just a representation of a linear transformation. Given that a matrix A corresponds to a linear transformation T, any matrix of the form B^{-1} A B (called the conjugate of A by B) for an invertible matrix B corresponds to the same transformation, represented in a difference basis. In particular, the eigen values of a matrix correspond to the eigen values of the linear transformation, so conjugating by an invertible matrix cannot change the eigen values.

It’s possible that you meant that you want to scale the eigen vectors so that each has unit length. This is a common thing to do since then the eigen values tell you how far a vector of unit length is magnified by the transformation.