Questions tagged [linear-algebra]

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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10
votes
1answer
197 views

Problems concerning subspaces of $M_{n}(\mathbb{Q}) $

Let $M_{n}(\mathbb{Q}) $ denote the $n$ times $n$ matrices over the rational number field. $N$ be a subspace of $M_{n}(\mathbb{Q}) $. Then if all the non-zero matrices in $N$ are invertible, what is ...
0
votes
0answers
17 views

Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere

Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
0
votes
0answers
35 views

Matrix irreducibility [closed]

Consider irreducible nonnegative matrix $\mathbf{A} \in \mathbf{M}_{n}(\mathbb{R})$ such that $a_{ij} \in [0,1)$ as element of $\mathbf{A}$ of period $p$. If $\mathbf{A}^{T}$ is transpose of $\mathbf{...
3
votes
1answer
97 views

On the equation $[U, V] - V_x = C(x)$

While considering the zero curvature equation $U_t - V_x + [U, V] = 0$, I developed a similar problem, albeit one that discards time dependence entirely. For a given $U(x)$ and $C(x)$, find $V(x)$ ...
2
votes
0answers
147 views

Hurwitz–Radon problem for $ \mathbb{Q} ^{n} $

What is the maximal number of orthogonal operators $ A _{1} , \dotsc, A _ {m} $ in $ \mathbb{Q}^{ n } $ satisfying the relations $ A_{i}^{2} = - I $ and $ A_{i}A_{j} + A_{j}A_{i} = 0 $ for $ i \neq ...
2
votes
0answers
25 views

Connection of the singular value before and after normalization

Given a matrix $P \in \mathbb{R}^{n \times d}$, we can get $P = U \Sigma V^T$ by using SVD. Let's say, we have another matrix $P' \in \mathbb{R}^{n \times d}$, it is the $P$ matrix with normalization ...
4
votes
1answer
232 views

Derivative of trace

Consider two positive-semi definite matrices $T_1, T_2$ of unit trace. Let $T(\lambda):=T_1 + \lambda(T_2-T_1)$ be the convex combination of the two. We then study $f(\lambda) := \operatorname{tr}(T(\...
1
vote
0answers
105 views
+50

Maximum mutual information of random unitary transformation

Let $\mathbf{U}$ and $\mathbf{V}$ be random unitary matrices independent of random input vector $\mathbf{x}$. Moreover, $\mathbf{z}$ be random iid complex Gaussian vector with zero mean and identity ...
2
votes
0answers
43 views

What rotations are used as a reduction step in Kenig-Ruiz-Sogge's uniform Sobolev estimate?

I think I have understood the bulk of the paper [KRS], but one of the parts I cannot understand is when the authors reduce Theorem 2.1 (p.332) into Proposition 2.1 (p.335). I can understand all the ...
2
votes
0answers
188 views

What is the name of this tensor?

A matrix M is usually called a hollow matrix if all of its diagonal elements are zero: $$ M_{pp} = 0, \quad \forall \: p. $$ We can generalize this to an $n$-way tensor T, such that: $$ T_{p_1 \cdots ...
12
votes
0answers
640 views

Seek for a algebro-geometric proof: the group homomorphism $\mathrm{SL}(2,\mathbb{Z}) \rightarrow \mathrm{SL}(2,\mathbb{Z}/N\mathbb{Z})$ is surjective

It is a well-known fact that the group homomorphism $\mathrm{SL}(2,\mathbb{Z}) \rightarrow \mathrm{SL}(2,\mathbb{Z}/N\mathbb{Z})$ is surjective. What I want is a proof by method of algebraic geometry. ...
0
votes
0answers
24 views

minimizing condition number of matrices

I am interested in the following problem: given a finite set of matrices $A_{n}\in GL_{k}(\mathbb{R})$, (actually- I am only interested in in k=2) find a matrix $M\in GL_{k}(\mathbb{R})$ that minimize ...
0
votes
0answers
52 views

Eigenvalue perturbation Problem

Consider a nonnegative matrix $\mathbb{K} \in \mathbf{M}_{n}(\mathbb{R}) $ with positive diagonal entries, which is perturbed by a small nonnegative matrix $\mathbb{E} \in \mathbf{M}_{n}(\mathbb{R}) $ ...
2
votes
1answer
64 views

Average the covariance matrix over all orthogonal matrices

Let $M=O\Lambda O^\top$ be a positive semi-definite matrix, where $\Lambda\in \mathbb{R}^{p\times p}$ is a diagonal matrix with non-negative entries and $O\in \mathbb{R}^{p\times p}$ is an orthogonal ...
1
vote
0answers
63 views

Linearly independent vectors from a matrix product

I have a product of $n$ integer-valued matrices: $$V=M_1 M_2 \dotsm M_n \,.$$ The $M_i$ are not square matrices, but the rows of $M_i$ and the columns of $M_{i+1}$ have the same length, so the ...
1
vote
0answers
25 views

Optimization problem on trace of complex matrix product

Given a complex rectangular matrix $A$ $(k \times n)$, I am interested in solving the following optimization problem over $(k\times n)$ complex matrices $x$: $$ \mathrm{arg}\max_X \,\mathrm{trace}(X^...
31
votes
8answers
3k views

Uncountable counterexamples in algebra

In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
7
votes
2answers
179 views

Is it possible for the reduction modulo $p$ of an non-commutative semisimple algebra to be commutative?

Suppose that $I, X_1, \ldots, X_{d-1}$ are $n \times n$ matrices with integer entries whose $\mathbb{Z}$-span is a subalgebra of $\mathrm{Mat}_n(\mathbb{Z})$. Suppose that, thought of as a subalgebra ...
0
votes
0answers
81 views

Large subgroups of infinite-dimensional vector spaces

Let $V$ be an infinite-dimensional vector space over $\mathbb{Q}$. Consider a proper subgroup $W$ of $V, +$ with the following property: each vector line $L$ (which we see as a subgroup of $V, +$) has ...
4
votes
1answer
123 views

Finding a binary variable assignment to make a matrix with variables singular (over F_p)

Consider a square matrix defined over a finite field $M\in\mathbb{F}_p^{n\times n}$ having the following form $$M=\begin{bmatrix}a_{11}+b_{11}x_1&a_{12}+b_{12}x_1&\dots&a_{1n}+b_{1n}x_1\\...
7
votes
0answers
74 views

Is this “semi-tensor product” something recently invented? Are there other usages of it?

The context: I was reading a paper in which they used the following definition called "Semi-Tensor Product" (STP) or "Cheng" product (In honor to its "inventor" D. Cheng):...
1
vote
2answers
235 views

Proving that a system of polynomial matrix equations over $\mathbb{F_2}$ has no solution

I am working on a problem involving nilpotent matrices over $\mathbb{F}_2$ and I was able to reduce it to proving that the system \begin{equation} \begin{cases} A^2+ BC+ BCA+ ABC+A = I_4 \\ ...
17
votes
3answers
665 views

Existence of translation-invariant basis on $C_c(\mathbb R)$

Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
1
vote
1answer
147 views

Prove or disprove a matrix inequality (positive semidefinite)

I want to prove or find a counterexample that there exist constants $\mu>0, \rho>0$ such that the following inequality holds: \begin{align} (H + \mu M)^2 \succeq \rho M^2, \end{align} where $\mu&...
1
vote
0answers
30 views

The rate of convergence of Markov chain to stationary distribution

Let $X_t$ is Markov chain with transition rates $c: G \times G \rightarrow [0: +\infty)$, where $c(x, y) > 0$, $c(x, x) = -\sum_y c(x, y)$ for $x \neq y$. If $\mu_t(x)$ is the distribution of chain ...
1
vote
1answer
73 views

Eigenvalues of a block matrix with zero diagonal blocks

Suppose $A$ is a $k_1\times k_2$ matrix with real entries, $k_1<k_2$. Let $M$ be the matrix \begin{equation} M:=\begin{pmatrix} 0_{k_1} & A\\ A^\top & 0_{k_2} \end{pmatrix}, \end{equation} ...
1
vote
0answers
51 views

What subspace of $\operatorname{SU}(4)$ group keeps an element of the $\mathfrak{su}(2)$ subalgebra within $\mathfrak{su}(2)$ upon adjoint action?

Consider the Lie group $G_4=\operatorname{SU}(4)$ with (15) generators $T^a$. A basis for the latter is $$\{\sigma^j \times 1_2, \quad \quad \sigma^i \times \sigma^j, \quad \quad 1_2 \times \sigma^j\},...
0
votes
0answers
21 views

Loss function for matrix with fixed trace

I have a matrix $P \in \mathbb{R}^{n \times m}$ where $n \gg m$ and each row of $P$ has norm $1$. It is not hard to see that the $P^T P$ has a fixed trace $n$. I am try to find a loss function to push ...
3
votes
2answers
103 views

Massive dirac operator symmetric spectrum

Consider the Dirac operator $$ H = \begin{pmatrix} m & -i\partial_z \\ -i\partial_{\bar z} & -m \end{pmatrix},$$ where $\partial_{\bar z}$ is the Cauchy-Riemann operator and $m \ge 0.$ It is ...
3
votes
1answer
166 views

Determinant of a matrix with entries specified by a set

Suppose $K$ is an $n\times n$ Hermitian matrix and $0\leq K\leq I$, which means that $I-K$ is positive semidefinite. Let $E\subset \{1,2,\dotsc,n\}$. I wonder how to show that det$(M^{E})\geq 0$, ...
-1
votes
0answers
57 views

Do there exist $P$ and $Q$ such that $PAQ$ is jordan canonical form of $A$? [migrated]

For $A \in M_n(\mathbb{F}_q)$, do there exist an invertible matrix $P$ and a monomial matrix $Q$ such that $PAQ = J$ is the Jordan canonical form of $A$?
0
votes
0answers
35 views

The linear embedding complexity of subsets of $0/1$ cube

We say $\pi$ is a subset of the $0/1$ integer points in $t$ dimensions represented by coordinates $(x_1,\dots,x_t)$ of complexity $\log^ct$ if there is an $A$ of $2^{\log^ct}\times2^{\log^ct}$ ...
0
votes
0answers
32 views

Fast decay of eigenvector elements

Let A be a set of similar (symmetric) matrices, sharing the same eigenvalues. I understand that their eigenvectors would be different. Let us focus on one eigenvector (e.g. corresponding to the lowest ...
1
vote
0answers
183 views

Solution to system of linear equations

Input: System of linear equations $$A[x_1,\dots,x_{t}]=b$$ where number of equations is at least number of variables but independence is not guaranteed. However there is atmost one non-negative ...
6
votes
1answer
165 views

Determinant of walk matrix for a skew-symmetric matrix of even order

Let $S=(s_{ij})$ be a skew-symmetric integral matrix of order $n$. We only consider the case that $n$ is even. Let $e$ be the all-one vector in $\mathbb{R}^n$. Define the walk matrix $$W(S)=[e,Se,\...
7
votes
1answer
306 views

Independent vectors in the permuting coordinates action of $S_n$ on $\mathbb{R}^n$

Let $V$ be the hyperplane in $\mathbb{R}^n$ with equation $\sum_i x_i=0$. The symmetric group $S_n$ acts on $V$ by $s\cdot (v_1,\ldots,v_n)=(v_{s^{-1}(1)},\ldots,v_{s^{-1}(n)})$. Consider those $v\in ...
0
votes
1answer
103 views

Linear independence of complex polynomials and a “sum of squares” conjecture

This will take me some time to explain. Let $n \geq 2$ be a fixed integer. Let $p_i(z)$, for $i = 1,\ldots,n$ be $n$ nonzero complex polynomials of degree at most $n-1$. I am interested in ...
3
votes
0answers
94 views

Geometry of elements with prescribed multiplicity eigenvalues

Let us take $G=\operatorname{Gl}(n,\mathbb{C})$ (considered as a linear algebraic group). Let us take $x \in G$: we know that its orbit $\mathcal{O}_x$ under the conjugation action is isomorphic (as ...
1
vote
1answer
38 views

Minimal volume of fundamental domains of lattices

Consider a full rank integer lattice in $\mathbb{R}^n$. Let $v_1$ be the shortest non-zero vector in the lattice, $v_2$ be the shortest one among those not parallel to $v_1$, $v_3$ be the shortest one ...
0
votes
0answers
16 views

Characterizing a variant of completely-Q matrices in linear complementarity problems

Given an $(n\times n)$-matrix $A$ (over the reals) and a vector $q \in R^n$, the linear complementarity problem $LCP(A,q)$ is the following problem: Find $w,z \in R^n_+$ such that $$w = q + Az,$$ and $...
2
votes
1answer
44 views

Derivative of eigenvalues of a symmetric tridiagonal matrix built via the Lanczos-Arnoldi scheme

Suppose $\mathbf{A}(\mu)$ being a symmetric positive definite matrix of dimension $n$ where its elements depend parametrically on the real parameter $\mu$. Suppose now to build the orthonormal basis ...
1
vote
1answer
52 views

What is the spectral norm of the matrix $\text{diag}(p)-pp^T$?

I have a probability vector $p$ s.t. $1^Tp=1$ and $p\geq 0$. I want to compute the spectral radius of the matrix $M=\text{diag}(p)-pp^T$ where $\text{diag}(p)$ has its diagonal elements as $p$ and its ...
0
votes
1answer
85 views

Are the eigenvalues of a symmetric matrix stay real after performing a row-addition operation? [closed]

Let $A \in M_n(\mathbb{R})$ be a symmetric matrix and assume we perform on $A$ an elementary row operation $A \xrightarrow{R_i = R_i + cR_j} B$ where $i \neq j$ and $c \in \mathbb{R}$ to get $B$. Are ...
4
votes
0answers
65 views

Subspaces of vanishing permanent

Suppose that $p\ge 5$ is a prime, $n$ a positive integer divisible by $p-1$, and $L<\mathbb F_p^n$ a subspace of dimension $d=n/(p-1)$. Do there exist vectors $l_1,\dotsc,l_n\in L$ such that the ...
0
votes
1answer
122 views

Localizing the intersections of cubics

For Hermitian matrices $A,B \in \mathbb{C}^{n \times n}$, can one readily compute a set of cones that separate the maxima of $$\frac{x'Ax}{x'Bx}$$ among $x$ with unit-norm components? i.e. where do ...
3
votes
1answer
76 views

Operator norm of difference of matrix decompositions

This question is in part related to a question that I have already posed. Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}_A \...
2
votes
1answer
66 views

Eigenvalues of large symmetric random tensors

I am studying the eigenvalues of large random tensors and realise that very little is known about it. I was wondering what is already known and what could be potential leads to find their limiting ...
3
votes
3answers
152 views

When is a linear subspace to be closed in all compatible topologies

Let $V$ be a real vectors space, and $W$ be a linear subspace. Say $W$ is obviously closed if, for every topology on $V$ that makes $V$ a Hausdorff locally convex topological vector space, the ...
3
votes
1answer
86 views

Eigenvalues of product of unitaries

Consider $d\times d$ unitary matrices $U, \, V, \, W$ such that $$ W=UV. $$ Suppose that the eigenvalues of $U$ and $V$ are $(e^{i\theta_1},\cdots,e^{i\theta_d})$ and $(e^{i\phi_1},\cdots,e^{i\phi_d})$...
0
votes
1answer
115 views

Topological characterization of invertible real matrices [closed]

Let $n\geq 2$ be an integer. Consider the topological space $M_n$ of $n$-by-$n$ matrices with real entries. Can you give a short non-constructive proof of the existence of a continuous function $M_n\...

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