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Concept# Diagonal matrix

Summary

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is \left[\begin{smallmatrix}
3 & 0 \
0 & 2 \end{smallmatrix}\right], while an example of a 3×3 diagonal matrix is
\left[\begin{smallmatrix}
6 & 0 & 0 \
0 & 0 & 0 \
0 & 0 & 0
\end{smallmatrix}\right]. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix.
A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values.
Definition
As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix D = (di,j) with n columns and n rows is diagonal if
\forall i,j \in {1, 2, \ldots, n}

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The purpose of the course is to introduce the basic notions of linear algebra and its applications.

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Since the 2008 Global Financial Crisis, the financial market has become more unpredictable than ever before, and it seems set to remain so in the forseeable future. This means an investor faces unprecedented risks, hence the increasing need for robust portfolio optimization to protect them against uncertainty, which is potentially devastating if unattended yet ignored in the classical Markowitz model, whose another deficiency is the absence of higher moments in its assumption of the distribution of asset returns. We establish an equivalence between the Markowitz model and the portfolio return value-at-risk optimization problem under multivariate normality of asset returns, so that we can add these excluded features into the former implicitly by incorporating them into the latter. We also provide a probabilistic smoothing spline approximation method and a deterministic model within the location-scale framework under elliptical distribution of the asset returns to solve the robust portfolio return value-at-risk optimization problem. In particular for the deterministic model, we introduce a novel eigendecomposition uncertainty set which lives in the positive definite space for the scale matrix without compromising on the computational complexity and conservativeness of the optimization problem, invent a method to determine the size of the involved uncertainty sets, test it out on real data, and explore its diversification properties. Although the value-at-risk has been the standard risk measure adopted by the banking and insurance industry since the early nineties, it has since attracted many criticisms, in particular from McNeil et al. (2005) and the Basel Committee on Banking Supervision in 2012, also known as Basel 3.5. Basel 4 even suggests a move away from the

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what-if" conditional value-at-risk' measure. We shall see that the former may be replaced with the latter or even other risk measures in our formulations easily.Consider a two-class classification problem where the number of features is much larger than the sample size. The features are masked by Gaussian noise with mean zero and covariance matrix Sigma, where the precision matrix Omega = Sigma(-1) is unknown but is presumably sparse. The useful features, also unknown, are sparse and each contributes weakly (i.e., rare and weak) to the classification decision. By obtaining a reasonably good estimate of Omega, we formulate the setting as a linear regression model. We propose a two-stage classification method where we first select features by the method of Innovated Thresholding (IT), and then use the retained features and Fisher's LDA for classification. In this approach, a crucial problem is how to set the threshold of IT. We approach this problem by adapting the recent innovation of Higher Criticism Thresholding (HCT). We find that when useful features are rare and weak, the limiting behavior of HCT is essentially just as good as the limiting behavior of ideal threshold, the threshold one would choose if the underlying distribution of the signals is known (if only). Somewhat surprisingly, when Omega is sufficiently sparse, its off-diagonal coordinates usually do not have a major influence over the classification decision. Compared to recent work in the case where Omega is the identity matrix [Proc. Natl. Acad. Sci. USA 105 (2008) 14790-14795; Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 367 (2009) 4449-4470], the current setting is much more general, which needs a new approach and much more sophisticated analysis. One key component of the analysis is the intimate relationship between HCT and Fisher's separation. Another key component is the tight large-deviation bounds for empirical processes for data with unconventional correlation structures, where graph theory on vertex coloring plays an important role.

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