# Tagged Questions

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### Are there some algorithms to solve the diagonal matrix $X$ to the following matrix equation?

Suppose $X$ is an unknown $m \times m$ diagonal matrix. Given a scalar $0<c<1$, and a matrix $A$ of $m \times m$ size whose entries $0<A_{i,j}<1$. Are there some algorithms to find the ...
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### Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation: $X=c \cdot AXA' - diag(c \cdot AXA')+ I$, where (1) $A \in R^{n \times n}$ is a given matrix whose element ...
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### Fast algorithm for maximizing smallest eigenvalue of linear combination of hermitian matrices

I have an engineering back ground. Due to work, I came across this problem \begin{align} &\max_{\lambda,y_i\in \mathbb{R}}~\lambda \\\ ...
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### eigen-decomposition solution? is it unique?

Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
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### Matrix optimization problem

This is (probably) an easy one: Given a positive definite matrix $M$, find the positive definite matrix $X$, which minimizes $\textrm{tr}(X M)$ subject to $\det(X) = 1$. Looking for how to find X, ...
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### efficient way to compute the inversion of the following matrix

Hi, there I have looked it up in the current textbook. The conventional numerical method to compute the inversion of an $n \times n$ matrix requires $O(n^3)$. However, for the following special ...
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### Finding the solution to b = Ax that minimizes the Hamming weight (everything over the field F_2).

Is there an efficient algorithm for finding the solution $x$ of $b = Ax$ that minimizes the Hamming weight of $x$, where $A$ is a nxm-matrix over the field $\mathbb{F}_2$ ("integer matrix modulo ...
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### Minimizing quadratic form over permutations

Let $Q$ be an $n \times n$ real symmetric matrix and $x$ an $n \times 1$ real vector. Consider the following minimization problem: $\min_{\pi \in S_n} ~(\pi x)^{\rm T} Q (\pi x)$, where $S_n$ ...
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### Condition number for Ellipsoid method matrix

Hello, When using the ellipsoid method (for solving a linear program for example), the volume of the ellipsoid at each iteration is proven to decrease, and do so by at least a factor of $e^{1/2n}$. ...