# Questions tagged [matrix-theory]

Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.

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### A particular selection of rows in upper triangular matrices

Let $A$ be a strictly upper triangular $n\times n$ matrix whose entries are either 0 or 1 (diagonal entries are all 0) with the nullity $m<n$. Let us denote $R_j$ and $C_j$ with the rows and ...
1 vote
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### When can we separate two pairs in ${\mathbb H}_n$, although it is not a lattice?

Recall that a lattice is a partially ordered set $E$ for which any pair $a,b\in E$ admits a least upper bound and a greatest lower bound. Remark that given four elements $a_i,b_j$ ($j=1,2$), in order ...
152 views

### Matrix inequalities in series form

While studying the positivity of mechanical systems, we land on the following conjecture but don't know how to prove this. If a square matrix $A \in \mathbb{R}^{m\times m}$ satisfies both the two ...
287 views

### Almgren's regularity Theorem ; a simple example?

Let me remind Almgren's regularity Theorem: the singular set of area-minimizing surface has codimension at least $2$. I wish to share here a simple example in low dimension, although I don't know ...
412 views

### A potential new norm for matrices and Horn's inequalities

I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite ...
220 views

### Generating $\mathbf{PGL}_2(\mathbb{Z})$ and $\mathbf{PGL}_2(\mathbb{Q})$

It is well known that $\mathbf{PGL}_2(\mathbb{Z})$ is finitely generated, and that $\mathbf{PGL}_2(\mathbb{Q})$ isn't. My question is: what is a fast, natural way to see these properties without ...
48 views

### Rank of Coxeter matrix

Let $Q$ be a quiver and $\Phi_Q$ be the Coxeter matrix of $Q$. Then $\Phi_Q\pm I$ are full-rank?
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### Is there a geometric meaning to this algebraic observation about the Takagi decomposition?

Let $S$ be a complex symmetric matrix, i.e. $S = S^T$. The Takagi decomposition is the statement that $$S = UDU^T$$ where $U^* = U^{-1}$ and $D$ is a diagonal (nonnegative real) matrix. Here's an ...
131 views

### Solvability of a matrix exponential equation - generalized matrix logarithm

For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation $$G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) .$$ Basic ...
1 vote
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### Dynamical formulation of the 2-Wasserstein distance for *discrete* matrix-valued measures

TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures. Let $(X, d)$ ...
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1 vote
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### Solve linear matrix equation involving convolution

I am facing following equation: $$A * X + C \cdot X = D$$ with: $A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure, $X \in \mathbb{R}^{n \times n}$ the ...
1 vote
126 views

### eigenvalues of matrices (with positive entries)

I am reading an old paper by Kawpien and Pelczynski, Studia Math. 1970. It claims that singular values of a matrix (with positive entries? I am not sure) is given by $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....
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### Linear subspaces of $\mathrm{GL}_n(\mathbb{R})$ whose inverses are also linear subspaces

$\DeclareMathOperator\GL{GL}$We will call a subset $S \subset \GL_n(\mathbb{R})$ a linear subspace if it is of the form $S = S'\cap \GL_n(\mathbb{R})$ for some $S'\subset M_n(\mathbb{R})$ which is a ...
1 vote
168 views

### Schatten norm inequality

Let $A,B$ be two $n\times n$ matrices. Find a lower bound of the $p$-th Schatten norm $\|(A-B)(A-B)^\ast\|_{S_{p/2}}^{1/2}$ in terms of Schatten norm of $\|(AA^*+BB^*)\|_{S_q}$ for any relation ...
An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations  (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star ...