0
votes
0answers
17 views

Which matrix norm gives the minimal variation of eigenvalues?

This is a follow-up of this question. The original question is intentionally as general as possible, because I was interested in the most general possible answer. I am now trying to understand its ...
0
votes
0answers
90 views

Maximize the maximum Eigenvalue under a diagonally constrained matrix

Suppose we have $N\times N$ Hermitian matrix $\mathbf{A}$ I want to find the real $N\times N$ diagonal matrix $\mathbf{D}$ that maximizes the sum of the maximum Eigenvalues : $\mathbf{D}=\arg\max ...
0
votes
0answers
22 views

Find $\underset{\omega}{\min}$ $\underset{\beta \in \sigma(A)}{\max}$ $|\frac{\omega - \beta}{\omega + \beta}|$

As part of an algorithm for the solution of a linear system I'm trying to find $\omega > 0$, $\omega \in \mathbb{R}$ so that $$\underset{\beta \in \sigma(A)}{\max}|\frac{\omega - \beta}{\omega + ...
2
votes
2answers
71 views

Minimum of a quadratic form

If $\bf{A}$ is a real symmetric matrix, we know that it has orthogonal eigenvectors. Now, say we want to find a unit vector $\bf{n}$ that minimizes the form: $${\bf{n}}^T{\bf{A}}\ {\bf{n}}$$ How can ...
1
vote
0answers
68 views

Maximizing the product of first Eigenvalues of rank-1 hermitian matrices

Suppose we have $L$ complex vectors $\mathbf{a}_{l}$ with dimension $N\times 1$ I want to solve this optimization problem $\mathbf{x}_{\mathrm{opt}}=\arg ...
0
votes
0answers
20 views

Trust region sub-problem Explicit Formula [duplicate]

Consider the $2 \times 2$ trust region sub-problem. Given $Q$ symmetric $2 \times 2$, vector $\mathbf b$ and $\Delta > 0$, find $\mathbf x$ that minimizes $f(x)=\frac {1}{2} \mathbf x^T Q \mathbf x ...
0
votes
1answer
65 views

Trust region sub-problem with Jacobi Condition

Consider the $2 \times 2$ trust region sub-problem. Given $Q$ symmetric $2 \times 2$, vector $\mathbf b$ and $\Delta > 0$, find $\mathbf x$ that minimizes $f(x)=\frac {1}{2} \mathbf x^T Q \mathbf x ...
0
votes
1answer
78 views

Largest eigenvalue of symmetric matrix

I am trying to understand why the $\lambda_{\max}$ function is convex given an $n\,x\,n$ symmetric matrix, let's call it $A$. I know from elementary property of eigenvalues that all the eigenvalues of ...
0
votes
1answer
38 views

Sample Variance in Principle Components Analysis

I was reading this Why is the eigenvector of a covariance matrix equal to a principal component?. And in the top answer, the poster mentions that if the covariance matrix of the original data points ...
0
votes
0answers
17 views

Spectral Relaxation and Eigendecomposition

Suppose I have the following optimization problem: $\underset{\mathbf{x}}{\mathop{\max }}\,{{\mathbf{x}}^{T}}A{{\mathbf{x}}^{T}}$ s.t $\mathbf{x}\in {{\left\{ -1,1 \right\}}^{N}}$. One possible ...
1
vote
0answers
41 views

Minimization via eigendecomposition of Hadamard matrix products

Let $\boldsymbol{\mathcal{R}}$ and $\boldsymbol{\mathcal{M}}$ be $n\times n$ Hermitian matrices (which are known) and let $\boldsymbol{\mathcal{G}}$ be a rank one $n\times n$ (unknown) Hermitian ...
3
votes
2answers
179 views

Minimize $x^TAx$, subject to $||x||=1$. Show that ${x^*}^TAx^*$ is the smallest eigenvalue of $A$ in magnitude.

I'm solving constrained optimization exercises for preparing my final exam. I got stuck at this question. $$ \begin{array}{ll} \text{min} & \mathbf{x}^T\mathbf{A}\mathbf{x} \\ \text{s.t.} & ...
1
vote
0answers
33 views

Approximating maxmimal value of Rayleigh Quotient in a set by minimizing distance towards the largest eigenvector.

Is the solution of the following two problems equal? If no, under what circumstances they will be equal? $P_1: argmax_{x\in S,x^Tx=1} x^TAx$ $P_2: argmax_{x\in S,x^Tx=1} ||x-\bar{x} ||_2$, where ...
2
votes
0answers
91 views

Is there a generalization of eigenvalues/eigenvectors to $L^p$ for $p\neq2$?

Given symmetric positive definite $m\times m$ matrix $\Sigma$, $1 \le p,$ and constant $c > 0$, consider the solution to the optimization problem: $$ \min_{v : \left\|v\right\|_p \ge c} ...
1
vote
0answers
109 views

maximize an objective function with an infinite component

Suppose I have the following maximization problem: $\log\det(\alpha K_p)-c\alpha$ with respect to $\alpha$ with $c$ being a constant and $m$ being the dimension of $K_p$. Here, one of the eigenvalues ...
1
vote
1answer
274 views

Largest eigenvalue, semidefinite programming

The problem is t minimize the largest eigenvalue of a function of x. objective: $$ min \ \ \ \lambda_{max}(A(x))$$ where $$A(x) = A_0+x_1A_1+x_2A_2+...x_nA_n$$ and all $A$ is positive semidefinite. ...
1
vote
0answers
107 views

How to represent and handle eigenvalue constraint in optimization

For example consider the problem $\min_X f(X)$ s.t. $\lambda_i(X+A)=\lambda_i(B)$ for $i \in {1,...,N}$ where $A$ and $B$ are full rank N by N matrix, $\lambda_i(X)$ is the i-th eigenvalue of $X$ ...
2
votes
0answers
76 views

Spectral/ Eigen-Value solution with a linear constraint?

Is there a spectral or eigen-value solution to finding $X$ such that $Tr(CX^TMX)$ is minimum for a symmetric matrix $C$ and a p.s.d matrix $M$. Also there is a linear constraint on the minimization ...
1
vote
2answers
86 views

Spectral/Eigen-value solution?

Is there a spectral or eigen-value solution to finding $d$ vectors $x_1...x_n$ such that $ \sum_{i,j=1}^{d} C_{i,j} \cdot x_i^\top M x_j $ is minimized, with $C_{i,j}$ being a constant real-scalar ...
2
votes
1answer
307 views

Symmetrically make this matrix orthogonal, but don't you dare use the Frobenius norm…

I have read many of the questions already here in regards to the Frobenius norm, but they do not help me too much. My question is, why is the Frobenius norm not considered a 'proper' norm? In a ...
3
votes
1answer
117 views

Orthogonality, Maximization and Eigen-Solution

I Have read that for a matrix of reals $Y$ and a p.s.d matrix $B$ that the Maximum of $ f(Y)=Tr(Y^TBY)$ subject to $Y^TY = I$ is achieved when $span(Y)$ equals the span of the first $d$ ...
2
votes
0answers
455 views

Is there any relation between the principal eigenvalue of sub matrix and the original matrix?

I am wondering whether there is any relation between principal eigenvalue of sub matrix and the original matrix. In fact I am facing a problem which is to select $n$ rows and $n$ columns from the ...
1
vote
1answer
120 views

An interesting eigenvalue problem

Let $A\in\mathbb{R}^{d\times d}$ and $B\in\mathbb{R}^{d\times d}$ be two positive definite matrices. $k$ is a real coefficient. Suppose the largest eigenvalue of $A-kB$ is $\lambda_1$. Is it possible ...
1
vote
1answer
210 views

Is maximizing $\det A$ equivalent to minimizing $\mbox{tr} A^2$?

Question: $A\in\mathbb{R}^{n\times n}$ is a positive definite matrix with constant trace, i.e., $A>0$ and $\mbox{tr} A=k$. Let $\lambda_1\ge\cdots\ge\lambda_n>0$ be the eigenvalues of $A$. Can ...
3
votes
1answer
320 views

Minimizing the sum of the $k$ smallest elements of the diagonal of a matrix

I have a $N\times N$ symmetric positive semidefinite matrix $Q$, and am considering a class of symmetric positive definite matrices having all eigenvalues in a given bounded interval $[a, b]$. Is it ...