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Following the recent work of Jiang and Lin (Linear Algebra Appl. 585 (2020) 45--49), we present more results (bounds) on Harnack type inequalities for matrices in terms of majorization (i.e., in partial products) of eigenvalues and singular values. We discuss and compare the bounds derived through different ways. Jiang and Lins results imply Tungs version of Harnacks inequality (Proc. Amer. Math. Soc. 15 (1964) 375--381); our results %with simpler proofs are stronger and more general than Jiang and Lins. We also show some majorization inequalities concerning Cayley transforms. Some open problems on spectral norm and eigenvalues are proposed.
The purpose of this paper is two-fold: we present some matrix inequalities of log-majorization type for eigenvalues indexed by a sequence; we then apply our main theorem to generalize and improve the Hua-Marcus inequalities. Our results are stronger and more general than the existing ones.
We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non ne
Denote by $P_n$ the set of $ntimes n$ positive definite matrices. Let $D = D_1oplus dots oplus D_k$, where $D_1in P_{n_1}, dots, D_k in P_{n_k}$ with $n_1+cdots + n_k=n$. Partition $Cin P_n$ according to $(n_1, dots, n_k)$ so that $Diag C = C_1oplus
A priori, a posteriori, and mixed type upper bounds for the absolute change in Ritz values of self-adjoint matrices in terms of submajorization relations are obtained. Some of our results prove recent conjectures by Knyazev, Argentati, and Zhu, which
Let $T=begin{bmatrix} X &Y 0 & Zend{bmatrix}$ be an $n$-square matrix, where $X, Z$ are $r$-square and $(n-r)$-square, respectively. Among other determinantal inequalities, it is proved $det(I_n+T^*T)ge det(I_r+X^*X)cdot det(I_{n-r}+Z^*Z)$ with equality holds if and only if $Y=0$.