Matrix normalization with respect to constraint

Question

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

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

Answer 1

Perhaps I do not understand your question, but as he reads, this makes no sense to me.

A matrix is ​​simply a representation of a linear transformation. Given that the matrix A corresponds to a linear transformation T , any matrix of the form B^{-1} AB (called the conjugate to A on B ) for an invertible matrix B corresponds to the same transformation presented in a difference basis. In particular, the eigenvalues ​​of the matrix correspond to the eigenvalues ​​of the linear transformation, therefore, conjugation using an invertible matrix cannot change the eigenvalues.

Perhaps you meant that you want to scale your own eigenvectors so that each of them has a unit of length. This is a common thing, because then the eigenvalues ​​tell you how large the unit length vector increases during the conversion.