Matrix Mathematics: Theory, Facts, and Formulas

By Dennis S. Bernstein. 2009

1,650 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/f104ab59fc0ffd5b1f9cdd7c4e8ba04a55ddec06

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

Citations in the corpus

Robust Dynamic Operating Envelopes for DER Integration in Unbalanced Distribution Networks (2022, methodology): “It can be shown by combining the theorem of the arithmetic and geometric means [53, Fact 1.17.14] and Cauchy’s inequality [53, Fact 1.17.3] (cid:15) (cid:5) i ) 1 / 2 unless all x i are equal, say to a value x ∗ , in which case where we have used i | x i | = ( x ∗ ) n and (cid:13) x (cid:13) 22 = i…”
A diagnostic criterion for approximate factor structure (2016, methodology): “By the max-min Theorem, the r.h.s. is such that: max A:dim(A)=k+1 min a∈A:‖a‖=1 a′Σh ( 1 n ∑ i θiθ ′ i ) Σha = µk+1 ( Σh ( 1 n ∑ i θiθ ′ i ) Σh ) .”
Fully Distributed Event-Triggered Protocols for Linear Multiagent Networks (2018, background): “In light of the Young’s inequality [25], it is not difficult to obtain that”
Unified M-estimation of fixed-effects spatial dynamic models with short panels (2018, neutral): “See, e.g, Bernstein (2009).”
Data-Driven Dissipativity Analysis: Application of the Matrix S-Lemma (2021, background): “By Haynsworth’s inertia formula (see [34, Fact 6.5.5]), Let ν be the number of negative eigenvalues of Ψ .”
State estimation under false data injection attacks: Security analysis and system protection (2018, neutral): “Lemma 1: [3] For two matrices M,N ∈ Cn×n, det(MN) = det(M)det(N).”
MINRES-QLP: A Krylov Subspace Method for Indefinite or Singular Symmetric Systems (2010, neutral): “The minimum-length solution is x† = [ 2 4 3 2 ].”
Distributed Compressive CSIT Estimation and Feedback for FDD Multi-User Massive MIMO Systems (2014, neutral): “t: (38) in (34). Choosing K 1 as K 1 = K+K o 2 r and combining (33), (34), (37) and (38) together, Theorem 1 is proved. B. Proof of Lemma 6 First, we introduce the following properties (39)-(41) from [45] which will be frequently used in the proof: XH i= X J(H i)J+ X inJ( H i) inJ;8J i: (39) (I P J ”
Derivative-Free Model Reference Adaptive Control of a Generic Transport Model (2010, neutral): “(26) produces _ V e t ; ~ Wt e t Qe t 2e t PB 1 ~ W t T x t 2e t PBu0002̂T2 t x t 2e t PB 2 t x t tr ~ W t ~ W t ~ W t ~ W t ~ W t 1 1 ~ W t u0002̂ T 2 t u0002̂2 t 2 t 2 t 2 u0002̂ T 2 t 1 ~ W t 2 ~ W t 1 2 t 2 u0002̂ T 2 t 2 t (27) Next, consider the fact ab aa bb=4 ( > 0), which follows fromYoung’”
Designing Fully Distributed Consensus Protocols for Linear Multi-Agent Systems With Directed Graphs (2013, background): “ if and only if G has a directed spanning tree. Lemma 2 ([18], [19]): Consider a nonsingular M-matrixL. Let q = [q1,··· ,qN]T = L−11. Then, G ,diag(1/q1,··· ,1/qN) > 0 and GL +LTG > 0. Lemma 3 ([20]): (Young’s Inequality) if a and b are nonnegative real numbers and p and q are positive real nu”