Journal of Mathematics, Engineering and Computer Sciences (JMEC)
Volume 0, Numéro 0, Pages 31-34
2022-07-15
The Cauchy Interlace Theorem For Symmetrizable Matrices
Authors : Saïd Kouachi .
Abstract
Symmetrizable matrices are those which are symmetric when multi-plied 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 sub matrix (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 or at last we can consider this paper as another proof based on the celebrated Dodgson’s identity [4] . This extension is interesting by the fact that in the symmetric case, the Cauchy interlace theorem together with the Courant-Fischer mini-max theorem and Sylvester’s law ofinertia, 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 who are contingent on the symmetric condition may occur for this large class of not necessarily symmetric matrices and open the door for many applications in future studies. We recall that our techniques are based on the celebrated Dodgson’s identity [4].
Keywords
Symmetrizable matrices, The Cauchy interlace theorem, Eigenvalues.
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