## 数学代写|凸优化作业代写Convex Optimization代考|ELEC4631

2023年4月3日

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## 数学代写|凸优化作业代写Convex Optimization代考|Gram matrix and realizability

The lengths, distances, and angles can be expressed in terms of the Gram matrix associated with the vectors $a_1, \ldots, a_n$, given by
$$G=A^T A, \quad A=\left[\begin{array}{lll} a_1 & \cdots & a_n \end{array}\right]$$
so that $G_{i j}=a_i^T a_j$. The diagonal entries of $G$ are given by
$$G_{i i}=l_i^2, \quad i=1, \ldots, n$$
which (for now) we assume are known and fixed. The distance $d_{i j}$ between $a_i$ and $a_j$ is
\begin{aligned} d_{i j} & =\left|a_i-a_j\right|_2 \ & =\left(l_i^2+l_j^2-2 a_i^T a_j\right)^{1 / 2} \ & =\left(l_i^2+l_j^2-2 G_{i j}\right)^{1 / 2} \end{aligned}

Conversely, we can express $G_{i j}$ in terms of $d_{i j}$ as
$$G_{i j}=\frac{l_i^2+l_j^2-d_{i j}^2}{2},$$
which we note, for future reference, is an affine function of $d_{i j}^2$.
The correlation coefficient $\rho_{i j}$ between (nonzero) $a_i$ and $a_j$ is given by
$$\rho_{i j}=\frac{a_i^T a_j}{\left|a_i\right|_2\left|a_j\right|_2}=\frac{G_{i j}}{l_i l_j},$$
so that $G_{i j}=l_i l_j \rho_{i j}$ is a linear function of $\rho_{i j}$. The angle $\theta_{i j}$ between (nonzero) $a_i$ and $a_j$ is given by
$$\theta_{i j}=\cos ^{-1} \rho_{i j}=\cos ^{-1}\left(G_{i j} /\left(l_i l_j\right)\right)$$
where we take $\cos ^{-1} \rho \in[0, \pi]$. Thus, we have $G_{i j}=l_i l_j \cos \theta_{i j}$.
The lengths, distances, and angles are invariant under orthogonal transformations: If $Q \in \mathbf{R}^{n \times n}$ is orthogonal, then the set of vectors $Q a_i, \ldots, Q a_n$ has the same Gram matrix, and therefore the same lengths, distances, and angles.

## 数学代写|凸优化作业代写Convex Optimization代考|Problems involving angles only

Suppose we only care about the angles (or correlation coefficients) between the vectors, and do not specify the lengths or distances between them. In this case it is intuitively clear that we can simply assume the vectors $a_i$ have length $l_i=1$. This is easily verified: The Gram matrix has the form $G=\operatorname{diag}(l) C \operatorname{diag}(l)$, where $l$ is the vector of lengths, and $C$ is the correlation matrix, i.e., $C_{i j}=\cos \theta_{i j}$. It follows that if $G \succeq 0$ for any set of positive lengths, then $G \succeq 0$ for all sets of positive lengths, and in particular, this occurs if and only if $C \succeq 0$ (which is the same as assuming that all lengths are one). Thus, a set of angles $\theta_{i j} \in[0, \pi]$, $i, j=1, \ldots, n$ is realizable if and only if $C \succeq 0$, which is a linear matrix inequality in the correlation coefficients.

As an example, suppose we are given lower and upper bounds on some of the angles (which is equivalent to imposing lower and upper bounds on the correlation coefficients). We can then find the minimum and maximum possible value of some other angle, over all configurations, by solving two SDPs.

# 凸优化代写

## 数学代写|凸优化作业代写Convex Optimization代考|Gram matrix and realizability

$$G=A^T A, \quad A=\left[\begin{array}{lll} a_1 & \cdots & a_n \end{array}\right]$$

$$G_{i i}=l_i^2, \quad i=1, \ldots, n$$
(目前) 我们假设它是已知的和固定的。距离 $d_{i j}$ 之间 $a_i$ 和 $a_j$ 是
$$d_{i j}=\left|a_i-a_j\right|2 \quad=\left(l_i^2+l_j^2-2 a_i^T a_j\right)^{1 / 2}$$ 反之，我们可以表达 $G{i j}$ 按妱 $d_{i j}$ 作为
$$G_{i j}=\frac{l_i^2+l_j^2-d_{i j}^2}{2},$$

$$\rho_{i j}=\frac{a_i^T a_j}{\left|a_i\right|2\left|a_j\right|_2}=\frac{G{i j}}{l_i l_j}$$

$$\theta_{i j}=\cos ^{-1} \rho_{i j}=\cos ^{-1}\left(G_{i j} /\left(l_i l_j\right)\right)$$

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## MATLAB代写

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