Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory

By Dennis S. Bernstein. 2005

662 citations recorded by the source corpus.

How researchers see this paper

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

1. Liangjun Su
2. Cynthia Rudin
3. M. Pesaran
4. Xia Wang
5. Tianyu Pang
6. Peter C. B. Phillips
7. Jun Zhu
8. Yikui Zhang

Citations in the corpus

• Specification Test for Panel Data Models with Interactive Fixed Effects (2014, unknown): “…00εi°°2 , where we have repeatedly used the rotational property of the trace operator, the fact that tr (AB) ≤ μ1 (A) tr (B) (B.7) for any symmetric matrix A and p.s.d. matrix B (see, e.g., Bernstein, 2005, Proposition 8.4.13), and the fact that tr (AB) ≤ tr (A) tr (B) (B.8) for any two p.s.d.…”
• Weak and Strong Cross Section Dependence and Estimation of Large Panels (2011, unknown): “`=m+1 i`f`t + "it; (18) See Bernstein (2005), p.”
• Estimation of large dimensional factor models with an unknown number of breaks (2018, background): “For example, under certain strong mixing and moment conditions on the process {F 0 t eit, t ≥ 1}, A5(i) can be verified by a simple use of Bernstein inequality for strong mixing processes provided that N and T diverge to infinity at comparable rates. See Moon and Weidner (2015) for primitive conditi”
• Sieve Estimation of Time-Varying Panel Data Models With Latent Structures (2019, background): “REFERENCE Bernstein, D. S. (2005), Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory.”
• Optimal Exact-Regenerating Codes for Distributed Storage at the MSR and MBR Points via a Product-Matrix Construction (2010, background): “Theorem 2 (MBR Exact-Regeneration): In the code presented, exact-regeneration of any failed node can be achieved by connecting to any d of the ( n − 1) remaining nodes.”
• Forecasting hierarchical and grouped time series through trace minimization (2015, methodology): “Assuming that I + ( C −W12W −122 ) W22 ( C′ −W −122 W ′ 12 )( W11 −W12W −122 W ′ 12 )−1 is non-singular, Fact 2.14.19 of Bernstein (2005) gives, P ∗1 =W22(C ′ −W −122 W ′ 12)(W11 −W12W −1 22 W ′ 12) −1 × [ I + (C −W12W −122 )W22(C ′ −W −122 W ′ 12)(W11 −W12W −1 22 W ′ 12) −1 ]−1 Wickramasuriya, Atha”
• Hadamard, Khatri-Rao, Kronecker and Other Matrix Products (2008, background): “…(Henderson and Searle (1979, 1981)), is involved in Jacobian matrices and matrix calculus (Mathai (1997) and Magnus and Neudecker (1999)), and is very recently used by several authors (see e.g. Bernstein (2005), Neudecker (2006), Neudecker and Trenkler (2006b), and Schmidt and Trenkler (2006)).”
• Shrinkage estimation of common breaks in panel data models via adaptive group fused Lasso (2016, unknown): “…matrix (see, e.g., Proposition 8.4.13 in Bernstein 2005, p.275) and that max(|min ()| max ()) = kksp for any symmetric matrix (see, e.g., Fact 5.10.3 in Bernstein 2005, p.194), we have°° ¡Φ̄−1 −Φ−10 ¢°°2 = °°Φ̄−1 ¡Φ̄ −Φ0¢Φ−10 °°2 = tr © Φ̄−1 ¡ Φ̄ −Φ0 ¢ Φ−10 0 Φ −1 0 ¡ Φ̄ −Φ0 ¢ Φ̄−1 0 ª ≤ £min…”
• PORTFOLIO LIQUIDATION IN DARK POOLS IN CONTINUOUS TIME (2012, unknown): “, Bernstein [2005], Theorem 8.”