Questions tagged [linear-algebra]

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

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Solving an underdetermined system of linear equations among the rationals, in the neighbourhood of an approximate solution

I have a system of linear equations $Ax=b$. Extremely underdetermined, for concreteness $x \in \mathbb{R}^{17,000}, b \in \mathbb{Z}^{156}$. $A$ is sparse, integer, full rank. I have a very precise ...
Dániel Varga's user avatar
0 votes
1 answer
197 views

Perturbation of matrices

Let $A(t)$ be a symmetric $n\times n$ matrix that continuously depend on $t\in [0,1]$. Let $\lambda_1(t)$ stand for the smallest eigenvalue for $A(t)$. Question. Does there exist a Lebesgue measurable ...
Ali's user avatar
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1 vote
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Positive semidefinite maximum principle

Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Let $\mu$ be a Borel probability measure on $M_n(K)$ supported on a compact set $C$ of positive semidefinite matrices with $\mathbf{0}\not\in C$. ...
Joseph Van Name's user avatar
12 votes
2 answers
713 views

Concrete representation of coend in linear algebra

$\require{AMScd}$Teaching coend calculus to a PhD student led me to this "elementary" computation that I would like to perform explicitly. Consider the functor $F : (\mathbb N,\le)^\text{op}\...
fosco's user avatar
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Spectrum of the convolution of the Maxwell collision kernel with a distribution

Given the Maxwell collision kernel $A(z) = |z|^2I_d - z \otimes z$, where $I$ denotes the $d\times d$ identity matrix and $z\otimes z = zz^T$ is the outer product, it is easy to see that $A(z)$ has ...
Vasily Ilin's user avatar
12 votes
1 answer
885 views

Positive 4-form

Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$. Let $Q$ be a quadratic form on $W$. Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
Anton Petrunin's user avatar
2 votes
1 answer
135 views

How to do LU factorization efficiently based on the factorized result added with a low-rank matrix?

Suppose a square $n\times n$, dense matrix $A^{\text{old}}$ has been factorized into $L^{\text{old}}$ and $U^{\text{old}}$ components by performing a LU decomposition $A^{\text{old}} = L^{\text{old}}U^...
Alex Joe's user avatar
2 votes
0 answers
103 views

Two definitions for transverse $(p,p)$ form

Let $V$ be a complex vector space of dimension $n$ and let $V^*$ be its dual. Fix any integer $1\leq p\leq {n-1}.$ A $(p,p)$-form $\alpha\in\bigwedge^{p,p}V^*$ is said to be strictly weakly positive ...
Invariance's user avatar
2 votes
1 answer
141 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 $N$ ...
Tomer Milo's user avatar
2 votes
1 answer
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Orthogonal complements in exterior powers

I previously asked this on Mathematics Stack Exchange, to no result: Consider the standard induced inner product structure on $\wedge^k\mathbb{R}^d$ given by defining $$\langle u_1\wedge \cdots \wedge ...
Ian Morris's user avatar
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Is there a combinatorial interpretation for the change of basis matrix in the Frobenius normal form representation?

Let $G$ be a graph on $n$ vertices. Let $A$ be the adjacency matrix of $G$ (i.e., rows and columns of $A$ are indexed by vertices of $G$, and the $(v,w)$ entry of $A$ is $1$ if $(v,w)$ is an edge in $...
Naysh's user avatar
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A general theory of pairings

Bilinear forms and bilinear maps for vector spaces over a field are standard material for an introductory course in linear algebra. There are also text books for bilinear forms and related quadratic ...
Thomas Preu's user avatar
5 votes
1 answer
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Relation between row space and column space resp. null space and left null space over general rings

Let $R$ be a ring and $M\in\text{Mat}(R,m\times n)$ a matrix for $m,n\in\mathbb{N}$. What results are known about the relation between column space (cs, image) and row space (rs), resp. null space (...
Thomas Preu's user avatar
3 votes
1 answer
194 views

Do radially bounded sets form a bornology?

We call a subset $A$ in a real vector space $E$ radially bounded if it intersects every ray emanating from $0$ via a bounded set. It is easy to see that radially bounded sets in $E$ form a bornology, ...
erz's user avatar
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1 vote
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Matrix valued word embeddings for natural language processing

In natural language processing, an area of machine learning, one would like to represent words as objects that can easily be understood and manipulated using machine learning. A word embedding is a ...
Joseph Van Name's user avatar
15 votes
2 answers
1k views

Positive quadratic polynomial

Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$. Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$. Is it possible to find a polynomial $\tilde q$ ...
Anton Petrunin's user avatar
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151 views

How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?

The classical Euler gamma function can be defined by the integral \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0. \end{equation*} Its ...
qifeng618's user avatar
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On the inequality-integer system [closed]

I need to prove this inequality, but I do not have a good background in algebra, if you can guide me: We have: $$ p_1 + 2p_2 + \ldots +kp_k < q_1 + 2q_2 + \ldots +kq_k+(k+1)q_{k+1}+\ldots+tq_t $$ ...
BADJARA Mohamed el Amine's user avatar
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Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$

Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group $G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
ghc1997's user avatar
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When can the eigenvalues be constrained to the right half plane as a result of column permutations?

Suppose we have a real square matrix. Under what conditions is it possible to permute the columns of the matrix such that all eigenvalues of the resulting matrix have nonnegative real part?
Max Aifer's user avatar
7 votes
1 answer
596 views

Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?

I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already) Let $Q $ be a matrix in $ \operatorname{GL}(...
ghc1997's user avatar
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Comparing the minimum squared Mahalanobis norm of $n$ [orthonormal vs. i.i.d] random unit vectors

Let $Q$ diagonal with coefficients $q_1 \geq ... \geq q_n \geq 0$. Let $U = \left[ u_1 ~ ... ~ u_n \right]$ distributed uniformly on the set of $n \times n$ orthonormal matrices, and $\tilde{u}_1, ...,...
guillaume's user avatar
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23 views

How can one orthogonalize the pointwise sum of two orthogonal sets?

Let $n = 2k$, and suppose that $V = \{v_1, \cdots, v_k\}$ is an orthogonal set in $\mathbb{R}^n$. In other words, the vectors in set $V$ are pairwise orthogonal to each other. Now, consider a new set $...
ABB's user avatar
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0 votes
2 answers
117 views

Reshaping data vector into a matrix for deconvolution using a circulant matrix

Suppose we have a circulant matrix S made from pseudorandom binary sequence of length $N$ consisting of $0$'s or/and $1$'s. $1$ means that we can inject something for chemical analysis and $0$ means ...
AChem's user avatar
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2 votes
1 answer
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An inequality related to matrix trace

$$Tr(A|A^T U \Sigma U^T|) \leq Tr(AA^T U \Sigma U^T)$$ where $A \in \mathbb{R}^{m\times n}$ is a real rectangular matrix, $U \in \mathbb{R}^{n \times n}$ is a orthonormal matrix and $\Sigma$ is a ...
Wayne's user avatar
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Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$

EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect. Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
Libli's user avatar
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1 vote
1 answer
200 views

Linear system with matrix as a variable

I have the following two linear systems: $$\begin{bmatrix} u_{11} & u_{12} \end{bmatrix} A = 0$$ $$\begin{bmatrix} u_{21} & u_{22} \end{bmatrix} B = 0$$ Both $A,B$ are $2 \times 2$ matrices ...
NYG's user avatar
  • 119
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0 answers
101 views

Partition of p-groups and vector spaces

I work on partition of $p$-groups such that $G/Z(G)$ is elementary abelian. As you know, we can think of such a factor group as a vector space. Therefore it connect to a partition of vector space. ...
Maryam's user avatar
  • 1
0 votes
0 answers
69 views

Norm of matrix product sum

Given matrices $A_{n\times n}, B_{n\times m}, C_{m\times m}$ such that $A^iBC^{N-i}$ is matrix with all zeros except upper right element for all $i$ from $0$ to $N$, what can we say about Frobenius ...
D.Ult's user avatar
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0 answers
59 views

Optimal top-k column subset

Let $V$ be a set of vectors over $\mathbb{R}^l$, $l\ge 1$, $\pi_i(V)$ be the permutation of vectors in $V$ such that they are ordered by their $i$th component (descending) in order for $\pi_i(V)(\...
Eli Bixby's user avatar
  • 101
0 votes
1 answer
67 views

Multivariate random variable problem [closed]

I'm stuck on this problem: Let a random vector $X$ be given in $\mathbb{R^{10}}$ with the standard scalar product. It is known that $\mathbb{E}[XX^T] = 5I_{10}$, $I_{10}$ – identity matrix of order 10....
Andrew's user avatar
  • 69
2 votes
0 answers
228 views

Functional continuity of eigenvalues?

We have the following theorems! Corollary VI.1.6 [Bhatia: Matrix Analysis]: Let $a_j(t)$, where $1\leq j \leq n$ be continuous complex valued functions defined on an interval $I$. Then there exists ...
VSP's user avatar
  • 233
5 votes
0 answers
131 views

Rings over which free modules of a certain rank are reflexive (satisfy Specker's theorem)

Following this question about the case of $\mathbb{Z}_{(p)}$, I've got to ask what is known more generally about rings and dimensions for which Specker's theorem holds. Let me make the following ...
Gro-Tsen's user avatar
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0 votes
0 answers
38 views

Intersection of subspace of cyclical rotations with orthant

In an $N$-dimensional real Euclidian space, let an orthant be specified by a vector $\underline{x}_0 = \{x_1, x_2, \dots, x_N\}$ where the components $x_k$ are binary in the sense that $x_k = \pm 1$...
Andreas's user avatar
  • 101
3 votes
0 answers
117 views

Computing Grothendieck group of coherent sheaves of affine toric 3-fold from a simplicial cone

Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$. Let $X=\operatorname{Spec}(k[\sigma^{\vee}\cap\mathbb{Z}^3])$ be the ...
Boris's user avatar
  • 501
3 votes
0 answers
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Unitary operators with the same inner product as vectors

Suppose we have a set of real unit vectors $v_1,\ldots,v_m \in \mathbb{R}^n$. We can always find a set of unitary operators $U_1,\ldots,U_m$ acting on $\mathbb{C}^N$ (for $N$ that is possibly much ...
user3826143's user avatar
3 votes
0 answers
45 views

Which invertible linear maps preserve the set of Hurwitz stable matrices?

Let $V = M_n(\mathbb{R})$ be the set of all $n\times n$ matrices with real elements and $V_{-}$ be a subset of Hurwitz stable matrices, i.e. matrices such that all their eigenvalues have strictly ...
Fiktor's user avatar
  • 1,264
8 votes
0 answers
149 views

Random walk on matrix until singularity

Consider a random walk on matrices, where one starts with the matrix $M=I_n$ and at each step randomly chooses an entry of $M$ to increase by $1$. I’m interested in two things about this walk: What’s ...
TheBestMagician's user avatar
2 votes
0 answers
119 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}(\...
ghc1997's user avatar
  • 763
1 vote
2 answers
115 views

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 $...
algebroo's user avatar
  • 113
0 votes
0 answers
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Analytical proof of an equation that includes transcendental functions

Can anyone help me find an analytical proof for the following statement: for $$t \in ]0,1[\setminus\{t_0\},\quad t' \in ]0,1[\setminus\{t_0\},\quad t\neq t',\quad k_1,k_2 \in \mathbf{R}^*,\quad t_0 \...
Tim's user avatar
  • 9
1 vote
0 answers
140 views

General linear group in infinite dimensions

Let $V$ be a vector space over the field $k$. Upon assuming the Axiom of Choice, we know that $V$ has a well-defined dimension $N$, and hence a well-defined basis $B$. Suppose that $N$ is not finite. ...
THC's user avatar
  • 4,313
3 votes
1 answer
73 views

Detecting linear operator from actions of powers on subspace

Say I have a sequence of linear operators $A_1,...,A_n$ on a (real) vector space $V_1$. I suspect that there's a second vector space $V_2$, and an operator $A$ on $V_1\oplus V_2$, such that $A_i=(\...
Dror Speiser's user avatar
  • 4,563
2 votes
1 answer
153 views

Product of a vector by an inverse of Toeplitz matrix

It is well known that using fast Fourier transform it's possible to multiply a vector by a Toeplitz matrix $A \cdot v = w$ in $n\cdot\log(n)$ operations. I read somewhere that also the product of a ...
Enea Olati's user avatar
9 votes
2 answers
801 views

What norms can be "universally" defined on any real vector space with a fixed basis?

Let $V$ be a real vector space and let $B = (b_\lambda)_{\lambda \in \Lambda}$ be a basis. So every $v \in V$ can be written uniquely as a linear combination $$ v = c_{\lambda_1} b_{\lambda_1} + c_{\...
AlpinistKitten's user avatar
5 votes
0 answers
196 views

Difficulty of solving $Ax=b$ in terms of limiting spectral density of $A$?

Suppose $A$ is a random real-valued $n\times n$ matrix and we want to know the difficulty of solving $Ax=b$ when entries of $b$ are sampled IID from Normal$(0,1)$. Can we say anything about the ...
Yaroslav Bulatov's user avatar
0 votes
0 answers
70 views

Uniform distribution on pairs of unitary matrices

This question has two parts. In Part 1, I would like to know if the following distribution on pairs of $d$-dimensional unitary matrices has popped up in the literature: "Uniform distribution on ...
user3826143's user avatar
1 vote
0 answers
27 views

Are there any known lower complexity bounds on solving positive semidefinite or positive semidefinite feasibility problems?

I've been trying to attack the problem posted here, about quickly checking if a matrix has any positive semidefinite completions. I suspect that the answer to the question is "no", because ...
seed's user avatar
  • 111
5 votes
2 answers
494 views

Is there a name for this family of matrices?

Let $0<a_1<a_2<\cdots<a_n$ and let $A$ be the symmetric $n\times n$ matrix with ${ij}^\text{th}$ entry $A_{ij}=\min\{a_i,a_j\}$. For example, if $a_i=i$ for each $i\le n=5$ then $$A=\begin{...
bldavies's user avatar
3 votes
1 answer
263 views

Enumerating possible number of satisfied linear equations

Consider a system of linear equations of variable $x=(x_1,\cdots,x_n)$ where each $x_i\in\{ 0,1,\cdots,L-1 \}$. Clearly, there are $\frac{n(n-1)}{2}$ number of equations in the system. $$x_i-x_j=0, \ \...
tony's user avatar
  • 297