Trace operator

Trace operator

Ridge Regression and Principal Component Analysis

Published in Taylor Arnold, Michael Kane, Bryan W. Lewis, A Computational Approach to Statistical Learning, 2019

Taylor Arnold, Michael Kane, Bryan W. Lewis

Inside the trace operator we can cyclically rotate a matrix product so that, for example, tr(ABC) is equal to tr(BCA). Using this trick here the V matrices cancel and we find an explicit forbula for the total variation tr(Var(β^))=σ2·tr(Σ-2) $$ tr(Var(\widehat{\beta })) = \sigma ^{2} \cdot tr(\Sigma ^{{ - 2}} ) $$ =σ2·∑j1σj2 $$ = \sigma ^{2} \cdot \mathop \sum \limits_{j}^{{}} \frac{1}{{\sigma _{j}^{2} }} $$

Try-once-discard scheduling for stochastic networked control systems

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Journal Information

Published in International Journal of Control, 2019

Mohammad H. Mamduhi, Sandra Hirche

In this article, the Euclidean norm, conditional expectation, and conditional probability are denoted by ‖ · ‖2, , and , respectively. The trace operator is represented by tr( · ). The ceiling operator determines the smallest following integer of a real number and is denoted by ⌈ · ⌉. A random vector X chosen from the multivariate Gaussian distribution with mean vector μ and covariance matrix W > 0 is represented by . A sequence of random variables xk evolving in time is shown by {xk}k, for all k ≥ 0. Superscripts and subscripts for state vectors indicate the belonging sub-system, and the time instance, respectively. For matrices though, subscripts indicate the corresponding sub-systems, while a superscript denotes the matrix power. Finally, we define an augmented vector as (x, y) := [x⊤y⊤]⊤ to simplify the notation.

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