# Tagged Questions

Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level ...

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### When does this matrix have full rank?

Suppose $\mathbf{B}\in\left[0,1\right]^{T\times M}$ is a binary matrix, $\mathbf{B}_{i}$ is a column of $\mathbf{B}$, and $\mathbf{X}\in\mathbb{R}^{N\times T}$ is a matrix where the columns are ...
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### Is it always possible to “separate” the eigenvalues of an integer matrix?

Call a square matrix Galois-irreducible if all its eigenvalues are Galois conjugates of each other. Let $M$ be an integer $n\times n$ matrix which is not Galois-irreducible. Is it always ...
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### A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to $L_{n}$

We define $$L_{n}=\{A=(a_{ij})\in M_{n}(\mathbb{R})\mid \sum_{i=1}^{n} a_{ij}=0 \;\;\;\text{for every fixed j}\}$$ This is a Lie subalgebra of $M_{n}(\mathbb{R})$. A dynamic-geometric proof for ...
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### Least-squares solution of systems of Sylvester equations

The Sylvester equation $AX+XB=C$ has been studied quite a lot and there are known algorithms for solving it. But has the situation where (an over-determined) system of equations $A_{i}X+XB_{i}=C_{i}$ ...
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### On Knot Equivalence problem statement

How is the knot equivalence problem represented? By this I mean I am looking for an analogy that compares with graph equivalence. For graph equivalence, we have two graphs $G_1$ and $G_2$ with ...
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### Existence or construction of a sequence of orthogonal matrices with three properties

This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help .... Any pointers or suggestions are appreicated! ...
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### Sum of the absolute eigenvalues of A>=B

Kindly help me to prove/disprove the following statement. Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ (...
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### Kernel of skew-symmetric matrix of rank $n-1$ with $n$ odd: is this a known result?

When $n$ is odd, the kernel of a skew-symmetric matrix $M$ of size $n\times n$ and rank $n-1$ is the span of $v$, where $v$ is a vector whose $i$-th component is the Pfaffian of the matrix obtained by ...
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### Exchange determinant and integral of a matrix-valued function

Assume $A(x)=(a_{ij}(x))_{k\times k}$ is a Hermitian matrix function on some manifold $M$, is there any inequality relates the integral of its determinant $\int_M det(A)$ and the determinant of its ...
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### Have “sturdy squares” been studied before?

Over on PPCG I've just made a question that involves arranging the numbers 1 to 9 on a 3 by 3 grid such that every 2 by 2 subgrid has the same sum. I'm calling such 3 by 3 grids and their N by N ...
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### A $k \times n$ matrix with a lot of invertible $k \times k$ submatrices over $\mathbb{F}_2$

In the appendix of the paper by Tolhuizen ( http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=841182) there is a very fast and easy probabilistic proof that for $k=cn$ where $c \in [0,1]$ is fixed, ...
### For of a special case of $Ax>=b$, are there always integer solutions?
The input consists of two integers $n\geq 2$ and $K\geq 2$, and a vector of positive integers $b$ of size $n$. We assume that $\sum_i b_i$ is a multiple of $K$. The output is a matrix $A(n\times n)$ ...