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.
450
questions
2
votes
1
answer
162
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Continuous path of unitary matrices with prescribed first column?
Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$.
Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
- 519
0
votes
1
answer
59
views
Inequality for extremal values of product of Hermitian matrices
I am looking for a reference to verify the following inequality, where $X$ and $Y$ are Hermitian positive semidefinite matrices:
$$
\lambda_n(X^{1/2}YX^{1/2}) = \lambda_n(XY) \leq \lambda_n(X)\...
0
votes
0
answers
61
views
Interpretate a condition on Signed graph
In my research I came up with a condition on signed complete graph as follows:
Consider adjacency matrix of signed complete graph, i.e. symmetric, diagonal-free matrix with elements $A_{ij}=-1$ or $+1$...
- 333
2
votes
0
answers
135
views
What are the name and inverse of an interesting integer matrix?
It is practicable to compute the matrix inverses
\begin{align*}
\begin{pmatrix}
1 & 0 & 0 \\
1 & 1 & 1 \\
1 & 2 & 2^2 \\
\end{pmatrix}^{-1}
&=\begin{pmatrix}
1 & 0 &...
- 796
5
votes
2
answers
335
views
How expressive is $e^A$ in the sense of universal approximation?
For any real matrix $B\in\mathbb{R}^{n\times n},n\ge 2$ and precision $\varepsilon$, is there a real matrix $A\in\mathbb{R}^{n\times n}$ such that $\|e^A-B\|_F<\varepsilon$? ($F$ refers to ...
- 109
0
votes
0
answers
18
views
Find condition of X such that I-XSA is nonsingular, where $S$ is skew-symmetric and $A$ is symmetric, nonsingular
Given $I_n$ is the identity matrix, $A \in \mathbb{R}^{n,n}$ is symmetric and nonsingular, and $B\in \mathbb{R}^{n,n}$ is skew-symmetric.\
a) For which condition of a matrix $X \in \mathbb{R}^{n,n}$, ...
- 33
2
votes
0
answers
109
views
Decompose a rational matrix as an integer matrix and an inverse of integer matrix
Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...
- 763
1
vote
2
answers
106
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Methods to solve for a matrix whose entries satisfy certain properties
(This question is a repost of a deleted question I asked, because the previous version had several elements missing)
Setting
For fixed $N \in \mathbb{N}$, I wish to compute the entries of a matrix $...
- 113
1
vote
0
answers
107
views
Some kind of product of two 2d tensors to create a 3d tensor?
I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays):
given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
- 461
1
vote
1
answer
97
views
Minimal number of linearly dependent rank-1 projectors
What is the minimal number of linearly dependent rank-1 projectors $\vec v \vec v^t$ in dimension n, under the condition that every set of n column vectors $\vec v$ is linearly independent.
PS: the ...
- 1,149
0
votes
0
answers
70
views
Efficient method to determine minimum eigenvalue of $2 \times 2$ block diagonal matrix
Suppose $H$ is a $2 \times 2$ block-diagonal symmetric matrix in $\mathbb{R}^{2^N \times 2^N} $. That is
$$ H = \begin{pmatrix} A_1 & 0 & \cdots & 0\\ 0 & A_2 & \cdots & 0 \\
...
- 131
8
votes
2
answers
493
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Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices
My question is motivated by this one, but within real matrices instead of complex ones.
${\bf Sym}_n(\mathbb R)$ is a vector space of dimension $N=\frac{n(n+1)}2$. Equipped with the scalar product $\...
- 50.5k
0
votes
0
answers
40
views
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 ...
- 3,898
1
vote
0
answers
67
views
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 ...
- 50.5k
2
votes
1
answer
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 ...
- 33
9
votes
2
answers
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 ...
- 50.5k
4
votes
1
answer
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 ...
3
votes
2
answers
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 ...
- 4,015
0
votes
0
answers
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?
- 489
4
votes
0
answers
93
views
A question on products of linear combinations of complex matrices
Suppose that $A_1, \dots , A_n, B_1, \dots , B_n \in \Bbb C^{d \times d}$ and that, for every $x \in \mathbb{C}^n$, the following holds
$$\left( \sum_i x_i A_i \right) \left( \sum_i x_i A_i \right)^{...
- 69
0
votes
1
answer
162
views
Faulty algorithm for simultaneous diagonalization?
I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
- 103
2
votes
0
answers
120
views
Visualisation of general 3x3 matrices, with applications to the pedagogy of linear algebra?
I've got a method for visualising non-zero $2 \times 2$ real matrices (modulo non-zero scalar factor) using the fact that:
Nonnegative determinant matrices (modulo non-zero scalar factor) are in 1-to-...
- 4,742
0
votes
0
answers
96
views
On the exponentiation of a stochastic matrix where the exponent is a function of matrix size
In this question, I asked about any arbitrary stochastic matrix $A(n)$ of the particular form
$$A(n) = \begin{pmatrix} 1 & 0 & \cdots & 0\\ x_{21} & x_{22} & \cdots & 0\\ \...
- 271
0
votes
0
answers
33
views
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 ...
- 4,742
3
votes
0
answers
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 ...
- 1,036
1
vote
0
answers
69
views
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 $(X, d)$ ...
- 105
2
votes
1
answer
103
views
Primal identity in matrix semigroup
Given a finite set of matrix $\{M_1,M_2,\cdots,M_n\}\subseteq \mathbb{C}^{d\times d}$, we consider the semigroup generated by matrix product.
We call $s_1\cdots s_k$ an identity index if
$M_{s_1}M_{...
- 1,471
0
votes
0
answers
33
views
Testing a condition in linear algebra involving Krylov subspaces
Let $A \in \mathcal{M}_n(\mathbb{R})$ be a real-valued $n \times n$ matrix.
For $b \in \mathbb{R}^n$, I consider the Krylov subspace
$$K_A(b) = \operatorname{span} \{ b, A b, \dotsc, A^{n-1} b \}.$$
...
- 948
1
vote
1
answer
148
views
Third order matrix differential norm
Suppose we have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is at least three times differentiable. Clearly, there is a relationship between the symmetric trilinear form $$T_1=\nabla^3f(x),$$ and ...
- 39
3
votes
1
answer
124
views
On the bounds of the sum of the squares of spectral variation of two real symmetric matrices
Suppose $A$ and $B$ are two symmetric real $\{0,1,-1\}$ matrices of order $n$ with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues $\lambda_1\ge \lambda_2\ge \dotsb \ge \...
- 61
8
votes
7
answers
991
views
One observation of special type of square matrix exponentiation
I was studying the following type of matrices,
$$
A = \begin{pmatrix}
1 & x_{12} & \cdots &x_{1n}\\
0 & x_{22} & \cdots &x_{2n}\\
\vdots\\
0&\cdots&0&x_{nn}
\end{...
- 271
3
votes
1
answer
179
views
The proof of the invertibility of $\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$
Suppose that $n$ is even. Any suggestion/appraoch to prove that $S=\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$ is invertible?
- 3,898
3
votes
1
answer
102
views
Matrix inequality in a paper by Piccinini-Spagnolo
In the paper 'On the Holder continuity of solutions of second order elliptic equations in two variables' by Piccinini and Spagnolo, they prove the following estimate:
$$
\begin{array}{ll}
\left(\int_S ...
- 473
1
vote
0
answers
56
views
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 ...
- 11
1
vote
1
answer
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 $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....
- 597
4
votes
1
answer
262
views
Positive system of algebraic integers
Let $\mathbb{A}$ be the ring of algebraic integers. Consider a sequence $(d_i)_{i \in I}$, with $I$ a finite set and $d_i \in \mathbb{A} \cap \mathbb{R}_{\ge 1}$, such that $$d_i d_j = \sum_{k \in I} ...
- 25.2k
1
vote
1
answer
68
views
A question about the sign of quadratic forms on nonnegative vectors
Let $M$ be a real square matrix of order $n\ge 3$.
Assume that for every nonnegative vector $\textbf{z}\in \mathbb R^n$ which has at lease one zero entry we have $\textbf{z}^T M \textbf{z} \ge 0$.
Can ...
- 13
2
votes
0
answers
40
views
Robustness of largest singular vectors with respect to noise
I would like to find a result that shows that the largest right-singular vectors of a data matrix are in some sense robust with respect to low-variance noise perturbations. Specifically, let $X = U D ...
- 61
1
vote
0
answers
104
views
Convex matrix combination
We all know the notion of a convex combination like
$$\lambda x_1 \; + \; (1 - \lambda) x_2$$ for some $\lambda \in (0, 1)$.
However, I am trying to find literature where this concept has been ...
- 61
0
votes
1
answer
167
views
Are there zero entries in the eigenvector corresponding to a simple eigenvalue?
For a real symmetric matrix $M$ and a simple eigenvalue $\lambda$, under which conditions the corresponding eigenvector has no zero entries? Perhaps, this is unconditional and one can provide a proof?
- 1
1
vote
0
answers
71
views
Solving a block tridiagonal system with diagonal perturbations
Say we have a block tridiagonal matrix, $T \in \mathbb{R}^{NL \times NL}$, with constant off diagonals, $\mathbf{B} \in \mathbb{R}^{L\times L}$, given by
$$
T = \begin{bmatrix} \mathbf{A}_1 & \...
- 11
9
votes
1
answer
239
views
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 ...
- 1,958
1
vote
0
answers
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 ...
- 97
8
votes
0
answers
399
views
Semigroups of matrices closed under conjugate transposition
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 ...
- 471
0
votes
0
answers
132
views
Multiplying by Loewner-ordered matrices
Suppose $A$ and $B$ are symmetric positive (semi-)definite, and $A<B$ in Loewner order, meaning $B-A$ is positive (semi-)definite. Is it true that, for a symmetric positive-definite $C$, we have $...
- 93
2
votes
1
answer
231
views
Classification of congruent integer matrices
I am interested in the following question:
Let $A,B\in\text{Mat}(2n\times2n;\mathbb{Z})$ be two integer matrices with the property that $\text{det}(A-A^T)=1=\text{det}(B-B^T)$. Are there known ...
1
vote
1
answer
125
views
The number of invertible 4×4 circulant matrices over the ring Z
Is the number of invertible 4×4 circulant matrices over the ring of integers Z finite?
I am looking for a reference which discusses this case.
- 11
1
vote
0
answers
198
views
Majorization for singular values of the difference of two matrices: $|\sigma(A)-\sigma(B)| \prec_w \sigma(A-B)$?
For two vectors $x$ and $y$ in $\mathbb{R}^n$, recall that $y$ weakly majorizes
$x$, denoted by $x\prec_w y$, if the sum of the $k$ largest entries of $x$ is smaller than or equal to that of $y$ for ...
- 213
1
vote
0
answers
39
views
What is the complexity of the matrix multiplication closure for a given generating system?
Given a generating set of $k$ matrices $X = \{M_1, M_2, \ldots, M_k\}$, with $M_i\in \mathrm{Mat}(\mathbb{C},n)$, what is the worst case complexity for computing the algebraic closure w.r.t. matrix ...
- 21
1
vote
0
answers
122
views
Identities for the determinant of a matrix similar to $\det(A)=\exp\circ\operatorname{tr}\circ\log(A)$ for different matrix functionals
The identity for the determinant of $A$ in the title is well know in matrix analysis and comes from the Jacobi's formula. I am interested in the existence of nontrivial formulas like this one (they do ...
- 573