The Cauchy interlace theorem for symmetrizable matrices

TL;DR

This paper extends the Cauchy interlace theorem from symmetric matrices to the broader class of symmetrizable matrices, enabling new applications in eigenvalue analysis and algorithms beyond symmetric cases.

Contribution

The paper generalizes the Cauchy interlace theorem to symmetrizable matrices, broadening its applicability and potential for future research and practical applications.

Findings

Extended interlacing property to symmetrizable matrices

Potential applications in eigenvalue and singular value decompositions

Opened avenues for algorithms on non-symmetric matrices

Abstract

Symmetrizable matrices are those which are symmetric when multiplied by a diagonal matrix with positive entries. The Cauchy interlace theorem states that the eigenvalues of a real symmetric matrix interlace with those of any principal submatrix (obtained by deleting a row-column pair of the original matrix). In this paper we extend the Cauchy interlace theorem for symmetric matrices to this large class, called symmetrizable matrices. This extension is interesting by the fact that in the symmetric case, the Cauchy interlace theorem together with the Courant-Fischer minimax theorem and Sylvester's law of inertia, each one can be proven from the others and thus they are essentially equivalent. The first two theorems have important applications in the singular value and eigenvalue decompositions, the third is useful in the development and analysis of algorithms for the symmetric eigenvalue problem. Consequently various and several applications whom are contingent on the symmetric condition may occur for this large class of not necessary symmetric matrices and open the door for many applications in future studies. We note that our techniques are based on the celebrated Dodgson's identity.