The Schur Complement Interlacing Theorem

Some inequalities involving determinants, eigenvalues, and Schur complements in Euclidean Jordan algebras

“…It turns out that a similar result holds in any simple Euclidean Jordan algebra. Proof Corresponding to the given α i and β j , there exists a real symmetric positive semidefinite matrix H with k × k nonsingular submatrix A such that the eigenvalues of H and H/A are α i and β j , respectively, see [13]. Since real numbers are included in complex numbers/quaternions/octonions, we have the result in the case of matrix algebras (and also in algebras isomorphic to these).…”

“…In [13], Hu and Smith showed that converse of Theorem 6 holds for complex Hermitian matrices. It turns out that a similar result holds in any simple Euclidean Jordan algebra.…”

Some interlacing theorems on the schur complement

“…In [7, 61 Hu and Smith completely determined the relations between the eigenvalues of H and those of H/A by proving the following interesting theorem. The proof of the necessity part of Theorem 1.2 was done in [7], and a rather long proof of the sufficiency part was given in [6]. In this note, we give a short proof of Theorem 1.2, and then extend our technique to prove an interlacing theorem on the singular values of the Schur Complements of rectangular complex matrices.…”

On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls

“…It should be pointed out that it can be of interest to apply the results on interlacing Cauchy's theorem and some of its extensions (see e.g., [20][21][22]) to the stability of aggregation models based on dynamic systems formulated via differential, difference or hybrid differential/difference equations.…”