All Questions
921 questions with no upvoted or accepted answers
3
votes
0
answers
1k
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Matrix Operations Preserving Hurwitz Stability
I begin with terminology I use in the question. A real square matrix $A$ is
negative-stable if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) < 0$;
$\ast$-negative-stable if for ...
- 105
3
votes
0
answers
162
views
Does the following characterization of subgroups of $GL_2(\mathbb{F}_p)$ generalise?
Let $p$ be a prime number. By a Cartan subgroup of $GL_n(\mathbb{F}_p)$ I mean an absolutely semisimple maximal abelian subgroup.
When $n=2$, it is well-known* that, for $G \subset GL_n(\mathbb{F}_p)$...
- 2,827
3
votes
0
answers
220
views
Could SVD be used to optimize the partial inner-products?
Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with
$m-$dimensional coordinates in ...
- 131
3
votes
0
answers
242
views
criterion for deciding whether the product of a sequence of Givens rotations can reach the full special orthogonal group
By Givens' rotation $R(1,2;\theta)$ I mean a matrix which has the
$$\begin{pmatrix} \hphantom{-}\cos \theta &\sin \theta \cr -\sin \theta & \cos \theta \end{pmatrix}$$
$2 \times 2$ block at ...
- 4,466
3
votes
0
answers
528
views
A question about the generalized Lidskii-Wielandt inequality for matrices proved by Thompson and Freede
In 1971, Thomson and Freede generalized the Lidskii-Wielandt inequalites as follows (version for singular values)
Let $A$, $B$ be $n\times n$ Hermitian matrices. Suppose $\alpha_1\geq \alpha_2 \geq \...
- 227
3
votes
0
answers
359
views
Do Isometry Groups Tell Us How Difficult Norms are to Compute?
The question: Consider two norms N1 and N2 on the space of n-by-n complex matrices. N1 and N2 have the same isometry group and computing N1 is NP-HARD. Does it follow that computing N2 is NP-HARD as ...
- 6,351
3
votes
1
answer
791
views
Real part of eigenvalues and Laplacian
I am working on imaging and I am a bit puzzled by the behaviour of this matrix:
$$A:=\left(
\begin{array}{cccccc}
1 & 0 & 0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 &...
user136033
3
votes
1
answer
740
views
Finding an adjacency matrix whose cube's diagonal is equal to a given vector
How can I find all binary matrices $A$ such that $A^3$ is a non-negative, integer square matrix and
$$\mbox{diag}\left(A^3\right)=b$$
for some given vector $b$?
Is there a way to characterize all ...
- 503
2
votes
0
answers
101
views
An inequality related to Problem 10210 AMM 1992 No. 3
Problem. Let $A$ be a $N \times N$ real matrix whose $(i,j)$ entry is $a_{ij} \ge 0, \forall i, j$. Let $1$ denote $N\times 1$ all-ones vector. Prove that
$$N^2 1^\top A^\top A A^\top 1 \ge (1^\top A ...
- 1,053
2
votes
0
answers
67
views
Preserving invertibility with adding rows
Suppose I have two $m\times n$ matrices $A$ and $B$ such that an $m\times m$ submatrix of $A$ is invertible if and only if the corresponding $m \times m$ submatrix of $B$ is. Now let's say I append a ...
- 21
2
votes
0
answers
113
views
Numbers of positive terms in polynomials equal A069999
Let $a(n)$ be A069999 (i.e., number of possible dimensions for commutators of $n \times n$ matrices; it is independent of the field). OEIS states that no generating function is known.
Let $P(n,k)$ be ...
- 4,938
2
votes
0
answers
35
views
Limiting spectral distribution of a random matrix with specific structure
First, consider an $N \times N$ Hermitian random matrix $V$ from the Gaussian Unitary Ensemble (GUE). It is well known that the empirical spectral distribution of the GUE satisfies the semicircle law ...
- 21
2
votes
0
answers
71
views
Are the ranks of the following matrices given by these simple expressions?
The question itself is formulated in the title, so below I specify the matrices and expressions mentioned there. In case if this is something known or can be easily deduced from something known, this ...
- 321
2
votes
0
answers
68
views
Nonlinear random matrix equations
Let $C$ be a matrix; $v$ be a column vector;
$P$, $\Delta$ are random matrices;
$x$ is a random column vector.
$$Cvv^T - \mathbb{E}[P^Tx]v^T - \mathbb{E}[P^Tx v^T \Delta] + O(\Delta^2)= 0$$
$$C^TCv - ...
- 21
2
votes
0
answers
71
views
Lexicographically largest incidence matrix
I have simple algorithmic question, but I can't find any source where this algorithm is explained in details.
Let's assume that we have incidence (with 0 and 1 values) matrix of size $m\times n$. Let ...
- 501
2
votes
0
answers
160
views
An "almost" true inequality for Hermitian matrices
Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality:
$$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
- 593
2
votes
0
answers
78
views
Partitions of bent vectors
Let $H=\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\ 1 & -1\end{bmatrix}.$ Let $A^{\otimes N}$ denote the tensor product of the matrix $A$ with itself taken $N$ times. We say that a vector $v$ of ...
2
votes
0
answers
117
views
A very specific quotient of a determinantal variety
I'm interesting in knowing whether a certain variety defined by maximal minors is irreducible. The specific construction is as follows: let $n \geq 2$ and let $R = \mathbb{C}[a_1,b_1,c_1,d_1,e_1,f_1,...
- 185
2
votes
0
answers
148
views
Nilpotent polynomial matrices over $F_q$ - polynomial count variety ? ( Nilpotent cone for Hitchin-Gaudin like integrable system)
Context: Number of nilpotent $n\times n $ matrices over $F_q$ is $q^{n(n-1)}$ classical result due to Ph.Hall, M.Gerstenhaber (see very nice exposition by T.Leinster at n-cat-cafe/arxiv) which have ...
- 24.9k
2
votes
0
answers
85
views
Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 83
2
votes
0
answers
72
views
Gradient descent over the set of complex symmetric matrices
In the course of my research (somewhat related to compressive sensing), I am trying to determine a complex, symmetric matrix $L$ (i.e. $L = L^T$) through the following optimization formulation:
$$ \...
- 21
2
votes
0
answers
58
views
Relationship between the homology of two types of tensor products of $\mathbb{Z}/ 2 \mathbb{Z}$-graded objects?
Let's consider a $2$-periodic complex $F$ of free $R$-modules, which is just a $\mathbb{Z} / 2 \mathbb{Z}$-graded complex
$$F_1 \xrightarrow{d_1} F_0 \xrightarrow{d_0} F_1$$
(really the arrow $d_0$ ...
- 553
2
votes
0
answers
146
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 &...
- 1,091
2
votes
1
answer
200
views
Upper bound for the rank of a Gram-type matrix
Let $V = (v_{1},...,v_{N})$ and $W = (w_{1},...,w_{N})$ be 2 sets, each containing $N$ vectors from $\mathbb{R}^{n}$; i.e, $v_{j}, w_{j} \in \mathbb{R}^{n}$ for all $1 \leq j \leq N$. Assume that $N$ ...
- 93
2
votes
0
answers
114
views
What is the quantity $\sqrt{\frac{c^2+d^2}{a^2+b^2}}$ of a matrix with determinant one?
Suppose that $A \in \mathbb R^{2 \times 2}$ has determinant one,
$$
A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
$$
I'm currently working on a problem where I obtained a condition on the ...
- 173
2
votes
0
answers
137
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}(\...
- 823
2
votes
0
answers
51
views
What conclusions can I derive from this family of trace inequalities?
Problem. Let $n_1,\ldots,n_s,m_1,\ldots,m_s\ge 0$ be nonnegative integers and set $m := \sum_{i=1}^s m_i$ and $n := \sum_{i=1}^s n_i$. Let $\oplus$ be an operation on matrices which stacks them in a ...
- 283
2
votes
0
answers
203
views
Permutation similarity of matrices with many distinct entries
This is related to graph isomorphism.
Here matrices are square $n \times n$ with non-negative integer entries.
Two matrices $A,B$ are permutation similar if there exist
permutation matrix $P$ such ...
- 25.4k
2
votes
0
answers
90
views
decidability special case of column generation problem
I have the following problem:
Input: sub-spaces $V_1, \dots, V_d$ of $\mathbb{Z}^{d}$
Question: are there $v_i \in V_i$ such that the matrix $(v_1, \dots, v_d)$ has determinant $\pm 1$ (equivalently, ...
- 55
2
votes
0
answers
97
views
How to decompose a matrix over a ring $F[X_1,\ldots,X_k]$ as a product of two matrices
Let $F$ be a field. Assume any reasonable conditions if needed, such as $F=\mathbb R$, $F=\mathbb C$, $F$ is a finite field, or $F$ has a specific characteristic, etc. Let $C$ be an $n\times1$ matrix ...
- 131
2
votes
0
answers
56
views
any ideas on how to solve matrix equation like this $X A_i Y = B_i$
the objective function is like
$$\operatorname*{argmin}_{X,Y} = \sum_i \|X A_i Y - B_i\|_F^2$$, and $A_i$ is a diagonal matrix
I've tried gradient-descent, but as it turns out not well, I wonder if ...
- 21
2
votes
0
answers
121
views
Entropy of eigenvectors of a large matrix
My question pertains eigenvectors of matrices with somewhat evenly distributed entries.
Let $M$ be an $N \times N$ matrix with complex entries (think of $N$ as a large integer). You can assume that $M$...
- 2,696
2
votes
0
answers
88
views
Nuclear norm minimization of convolution matrix (circular matrix) with fast Fourier transform
I am reading a paper Recovery of Future Data via Convolution Nuclear Norm Minimization. Here, I know there is a definition for convolution matrix.
Given any vector $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)^...
- 21
2
votes
0
answers
233
views
Do you know this formula for the scalar product in barycentric coordinates?
I've found a formula for a scalar product in barycentric coordinates which I think is pretty cool. I hope that it's new. Is it?
Suppose that you have points $x_1,\dots,x_n$ sitting in general position ...
- 1,048
2
votes
0
answers
107
views
Gradient of QZ decomposition
Let $A$ and $B$ be an $m \times n$ matrix of rank $ k_1 \le \min(m,n) $ and $ k_2 \le \min(m,n) $. Then the QZ decomposition or the generalized Schur decomposition is $A = USV^T$ and $B = UTV^T $, ...
- 61
2
votes
0
answers
161
views
Jacobi formula for matrices: variations
Jacobi’s formula says: $\frac{d}{dt}\text{det}(A(t))=\text{det}(A(t)) \cdot \text{tr}(\text{Ad}(A(t))\cdot\frac{d}{dt}(A(t))$.
Exists maybe a variation of the Jacobi’s formula where $\text{det}(\frac{...
- 83
2
votes
0
answers
69
views
Unimodular eigenvalue of a H-self-adjoint matrix (indefinite innerproduct)
Let $A,H \in \mathbb{C}^{n \times n}$ be such that $H$ is Hermitian and invertible and $A = H^{-1} A^* H$. In this case, $A$ is said to be $H$-self-adjoint. This is due to the fact that if $\langle \...
- 175
2
votes
0
answers
97
views
Fractional reverse direction Cauchy-Schwarz inequality
If $Z_1,\dots,Z_r$ are complex $m\times m$-matrices, then let $\Phi(A_1,\dots,A_r):M_m(\mathbb{C})\rightarrow M_m(\mathbb{C})$ be the linear mapping defined by $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\dots+...
2
votes
0
answers
154
views
Applying 1D integral to matrix integral
In the proof for finding an analytic solution to the propagation of a Hermite-Gaussian beam though a paraxial system given in the paper "The elliptical Hermite–Gaussian beam and its propagation ...
- 83
2
votes
0
answers
118
views
A gsvd variation: Two SVDs with common matrix
The generalized singular value decomposition (gsvd) is described on wikipedia here. Since there are several conventions, I'll just briefly present one.
The "MATLAB" convention decomposes two ...
- 61
2
votes
0
answers
538
views
Eigenvalues of the sum of matrices, where matrices are tensor products of Pauli matrices
recently I've been studying the toric code (a squared lattice in the context of quantum computation). I want to calculate the energy of the ground state and of all the excitations, with the respective ...
- 21
2
votes
0
answers
121
views
Eigenvalues of two positive-definite Toeplitz matrices
Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are:
$$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
- 21
2
votes
0
answers
57
views
spilt the sum of singular values of matrices
Let $A_{i} \in GL(d, \mathbb{R})$ for $i=1, 2, 3.$ For $q>0$, we denote $t_{3}^{q}=\sum_{i=0}^{3} \sigma_{1}^{q}(A_{i})\sigma_{2}^{q}(A_{i})\sigma_{3}^{q}(A_{i})$, $t_{2}^{q}=\sum_{i=0}^{3} \sigma_{...
- 1,043
2
votes
1
answer
506
views
Effect of duplicated row on singular values and vectors
Let $\mathbf{A}$ be a $n\times n$ matrix with Singular Value Decomposition (SVD) $\mathbf{A}=\mathbf{U}\mathbf{S}\mathbf{V}$ and $\mathbf{a}_1$ be the first row of $\mathbf{A}$. What can we say about ...
- 287
2
votes
0
answers
72
views
How to delete the maximum number of rows of a Boolean matrix by maintaining the sum greater than zero in each column
I have a Boolean matrix (entries are "0" or "1"), which is not square. The sum over each row and each column is constant (but they can be two different values). I would like to ...
- 21
2
votes
0
answers
152
views
A truncated Frobenius norm of a matrix is convex or not?
Given a positive integer $k$ and a matrix $X\in \mathbb{R}^{m\times n}$. A truncated frobenius norm of a matrix $X$ is defined by
$$\Vert X \Vert_{k,F} = \sqrt{\sum_{i=k+1}^{m} \sigma_i^2(X)},$$
where ...
- 143
2
votes
0
answers
345
views
Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues
In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
- 145
2
votes
0
answers
103
views
Generalisations of Volodin's construction of algebraic K-theory
In a previous question I asked about uses of Volodin's construction of the algebraic K-theory of rings. Some of these are striking and it made me wonder whether those proofs can be extended. This ...
- 637
2
votes
0
answers
504
views
Finding a basis for the range of a linear function
I realize this question is not high level but I have posted it on Math Stackexchange:
Stackexchange question
and have received some upvotes but no answers or comments, so I am trying here.
I will need ...
- 121
2
votes
0
answers
116
views
Intuition behind eigenvalues of a graph mattering
Is there a good intuition behind why the eigenvalues of a matrix corresponding to a graph tell us useful information about the graph? There are a lot of results relating the eigenvalues of the ...
- 6,969