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The (real) Grothendieck constant 
${K}_{G} $ is the infimum over those 
$K\in (0, \infty )$ such that for every 
$m, n\in \mathbb{N} $ and every 
$m\times n$ real matrix 
$({a}_{ij} )$ we have The classical Grothendieck inequality asserts the nonobvious fact that the above inequality does hold true for some 
$$\begin{eqnarray*}\displaystyle \max _{\{ x_{i}\} _{i= 1}^{m} , \{ {y}_{j} \} _{j= 1}^{n} \subseteq {S}^{n+ m- 1} }\sum _{i= 1}^{m} \sum _{j= 1}^{n} {a}_{ij} \langle {x}_{i} , {y}_{j} \rangle \leqslant K\max _{\{ \varepsilon _{i}\} _{i= 1}^{m} , \{ {\delta }_{j} \} _{j= 1}^{n} \subseteq \{ - 1, 1\} }\sum _{i= 1}^{m} \sum _{j= 1}^{n} {a}_{ij} {\varepsilon }_{i} {\delta }_{j} . &&\displaystyle\end{eqnarray*}$$
$K\in (0, \infty )$ that is independent of 
$m, n$ and 
$({a}_{ij} )$. Since Grothendieck’s 1953 discovery of this powerful theorem, it has found numerous applications in a variety of areas, but, despite attracting a lot of attention, the exact value of the Grothendieck constant 
${K}_{G} $ remains a mystery. The last progress on this problem was in 1977, when Krivine proved that 
${K}_{G} \leqslant \pi / 2\log (1+ \sqrt{2} )$ and conjectured that his bound is optimal. Krivine’s conjecture has been restated repeatedly since 1977, resulting in focusing the subsequent research on the search for examples of matrices 
$({a}_{ij} )$ which exhibit (asymptotically, as 
$m, n\rightarrow \infty $) a lower bound on 
${K}_{G} $ that matches Krivine’s bound. Here, we obtain an improved Grothendieck inequality that holds for all matrices 
$({a}_{ij} )$ and yields a bound 
${K}_{G} \lt \pi / 2\log (1+ \sqrt{2} )- {\varepsilon }_{0} $ for some effective constant 
${\varepsilon }_{0} \gt 0$. Other than disproving Krivine’s conjecture, and along the way also disproving an intermediate conjecture of König that was made in 2000 as a step towards Krivine’s conjecture, our main contribution is conceptual: despite dealing with a binary rounding problem, random two-dimensional projections, when combined with a careful partition of 
${ \mathbb{R} }^{2} $ in order to round the projected vectors to values in 
$\{ - 1, 1\} $, perform better than the ubiquitous random hyperplane technique. By establishing the usefulness of higher-dimensional rounding schemes, this fact has consequences in approximation algorithms. Specifically, it yields the best known polynomial-time approximation algorithm for the Frieze–Kannan Cut Norm problem, a generic and well-studied optimization problem with many applications.
Let 
${ \mathbb{K} }^{m\times n} $ denote the set of all 
$m\times n$ matrices over a skew field 
$ \mathbb{K} $. In this paper, we give a necessary and sufficient condition for the existence of the group inverse of 
$P+ Q$ and its representation under the condition 
$PQ= 0$, where 
$P, Q\in { \mathbb{K} }^{n\times n} $. In addition, in view of the natural characters of block matrices, we give the existence and representation for the group inverse of 
$P+ Q$ and 
$P+ Q+ R$ under some conditions, where 
$P, Q, R\in { \mathbb{K} }^{n\times n} $.
We prove a normalized version of the restricted invertibility principle obtained by Spielman and Srivastava in [An elementary proof of the restricted invertibility theorem. Israel J. Math. 190 (2012), 83–91]. Applying this result, we get a new proof of the proportional Dvoretzky–Rogers factorization theorem recovering the best current estimate in the symmetric setting while we improve the best known result in the non-symmetric case. As a consequence, we slightly improve the estimate for the Banach–Mazur distance to the cube: the distance of every 
$n$-dimensional normed space from 
${ \ell }_{\infty }^{n} $ is at most 
$\mathop{(2n)}\nolimits ^{5/ 6} $. Finally, using tools from the work of Batson et al in [Twice-Ramanujan sparsifiers. In STOC’09 – Proceedings of the 2009 ACM International Symposium on Theory of Computing, ACM (New York, 2009), 255–262], we give a new proof for a theorem of Kashin and Tzafriri [Some remarks on the restriction of operators to coordinate subspaces. Preprint, 1993] on the norm of restricted matrices.
Let 
$A, B$ be two square complex matrices of the same dimension 
$n\leq 3$. We show that the following conditions are equivalent. (i) There exists a finite subset 
$U\subset { \mathbb{N} }_{\geq 2} $ such that for every 
$t\in \mathbb{N} \setminus U$, 
$\exp (tA+ B)= \exp (tA)\exp (B)= \exp (B)\exp (tA)$. (ii) The pair 
$(A, B)$ has property L of Motzkin and Taussky and 
$\exp (A+ B)= \exp (A)\exp (B)= \exp (B)\exp (A)$. We also characterise the pairs of real matrices 
$(A, B)$ of dimension three, that satisfy the previous conditions.
Trace inequalities for sums and products of matrices are presented. Relations between the given inequalities and earlier results are discussed. Among other inequalities, it is shown that if A and B are positive semidefinite matrices then 
for any positive integer k.
