# Tagged Questions

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### What are the upper bound and stability conditions for the following simple linear system?

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1)$$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
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### Multi-dimensional permanent of structured tensor

I am facing the multidimensional permanent $$\text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j }$$ of a 3-tensor $W_{j,k,l}$ of ...
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### Solution of infinite dimension linear system

Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$. For fix n, we can construct n dimension linear equation ...
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### Cholesky decomposition of a large covariance matrix

I have a tricky problem concerning a covariance matrix cholesky decomposition. What I need is to obtain the cholesky decomposition of the estimated variance matrix of the set of samples stored in a ...
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### A variant of Cholesky decomposition involving binary matrices

Studying a problem that is not directly related to linear algebra I came across the following problem. Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...
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### Any generic way to move a psd matrix to its neighbors?

Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, ...
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### how to find all the solutions to $I+A+\cdots+A^n=0.$ [closed]

Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying ...
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### Perturbation of Cholesky decomposition for matrix inversion

I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$ where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...
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### A Problem on Linear Algebra

I'm trying to calculate an integral over the generalized Poincare upper half plane, then I find that I need to show the following identity: Let $X=(X_{i,j})\in\mathrm{GL}(n,\mathbb R)(n\geq 3)$ ...
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### What is the significance of matrix ordered algebras?

I am trying to grok matrix ordered operator algebras, but I am having a hard time understanding their significance from the definition. Here is the definition (or at least, one way of stating it): ...
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### On the linear transformation between matrix space

Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix. Suppose there exists ...
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### Possible diagonal values of a product of matrices with some specific characteristics

Hello all, This is a question that might or might not be related to my previous one. Imagine you have two matrices: Matrix $\mathbf{\Phi}=[\Phi_1,\ldots,\Phi_M]\in\mathbb{R}^{L\times M}$ where ...