Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,873 questions
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On the determinant of a class symmetric matrices
Consider the matrix $2\times2$ symmetric matrix:
$$
A_2=\begin{pmatrix} 1 & a_1 \\ a_1 & 1\end{pmatrix}.
$$
It's clear that the restriction $|a_1|<1$ implies that $\det(A_2)>0$. Moreover,...
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Free subgroups of $\mathrm{GL}(2,\mathbb{Z})$
Is there a bound $B$ such that every 2-generator subgroup
$G = \langle a, b \rangle \le {\rm GL}(2,\mathbb{Z})$
whose generators do not satisfy a relation of length $\leq B$ is free?
If it exists, ...
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Representation of vectors in $\mathbb{R}^2$ via differences of small vectors.
Is the following fact true?
Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k \...
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How to show a certain determinant is non-zero
For any $n$ distinct points $x_1,x_2 , \ldots , x_n$ on the real line show that
the matrix $M$ where $M(i,j) = e^{\lambda_j x_i} $ has non-zero determinant
where $\lambda_1 \lt \lambda_2 \lt \ldots \...
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Maximum dimension of space of matrices with a real eigenvalue
Let $M_n(\mathbb{R})$ denote the space of all $n\times n$ real
matrices. What is the maximum dimension $f(n)$ of a subspace $V$ of
$M_n(\mathbb{R})$ such that every matrix in $V$ has at least one real
...
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Pairs of matrices for which traces of powers are independent of the order
Let $A,B$ be $n\times n$ matrices over ${\mathbb C}$ such that, for all $m,k$ and all partitions $(i_1,\ldots ,i_r)$ of $m$ and $(j_1,\ldots ,j_r)$ of $k$ (perhaps with some zero parts),
$${\rm tr}\, (...
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The geometry of the solution set of a symmetric equation in four symmetric matrices
Let $n$ be a natural number. We can view the space of invertible symmetric matrices over a field as an open subset of$\mathbb A^{(n^2+n)/2}$. Inside the fourth power of this space, we have the closed ...
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Necessary and sufficient conditions for a sum of idempotents to be idempotent
Given: a finite list of $n$-by-$n$ idempotent complex matrices $E_1, E_2, \ldots, E_k$.
If all pairwise products $E_i E_j$ (with $i \neq j$) are zero, it is trivial to show the sum $E_1 + E_2 + \cdots ...
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Invariants and orbits of $n$-tensors
My question may be absolutely elementary and is probably answered in 19th century. A reference or a short clear argument would be highly appreciated.
Let $V_1, \ldots V_n$ be finite dimensional ...
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Symbols of elliptic operators
First let me state the problem, then I'll explain its origin and finally, I'll ask the main question..
Problem S. Fix a positive integer $n$. Find all the pairs $(V, S)$, whith the following ...
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Submodules of $({\mathbb Z}/6{\mathbb Z})^n$ intersecting $\{0,1\}^n$ trivially
$\newcommand{\F}{{\mathbb F}}$
$\newcommand{\Z}{{\mathbb Z}}$
Suppose that $\F$ is a finite field of prime order $p:=|\F|$, and let $n$ be a positive integer. I consider the regime where $\F$ is ...
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On minimal eigenvalue
Is it true that $\min\left(\lambda_{\min}(M_{12}),\lambda_{\min}(M_{13}),\lambda_{\min}(M_{23})\right) \le \frac{7}{20}$ where $M_{ij}$ is the matrix obtained by selecting the entries at the ...
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Questions on Toeplitz matrices: invertibility, determinant, positive-definiteness
These questions are probably very basic but I'll dare to ask them anyway since I didn't have much luck in Math Stack Exchange.
Let $A$ be an $n \times n$ Hermitian Toeplitz matrix:
$$A = \begin{...
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Which quadratic forms on $\Lambda^2 V$ come from quadratic forms on $V$?
Let $V$ be a finite dimensional vector space, say over $\mathbf R$. Let $g \in S^2 V^*$ be a quadratic form on $V$. Then $g$ induces a quadratic form $\Lambda^2 g \in S^2 \Lambda^2 V^*$ on $\Lambda^...
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What are the periodic Dyck paths?
I changed the thread completely so that everything is now elementary linear algebra.
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
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How to compute the rank of a matrix?
Okay, that's a misleading title. This is a somewhat subtler problem than undergraduate linear algebra, although I suspect there's still an easy answer. But I couldn't resist :D.
Here's the actual ...
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Conceptual explanation for curious linear-algebra fact in characteristic $2$
All matrices and vectors in this post have entries in the field $\mathbb{F}_2$.
Fix some $n \geq 1$. For an $n \times n$ matrix $X$, write $X_0$ for the column vector whose entries are the diagonal ...
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Reconstructing a word
Let $w(a,b)$ be a word in two letter alphabet. Let $$A=\left(\begin{array}{lll}x_1 & x_2 & x_3\\\ x_4 &x_5 & x_6\\\ x_7 & x_8 & x_9\end{array}\right), B=\left(\begin{array}{lll}...
user6976
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Matrix with small elements and prescribed determinant
Let $p$ be a large prime number. I want a $k\times k$ matrix with determinant $p$ and bounded integer elements (say, from -100 to 100). For which minimal $k$ such a matrix does always exist? We can ...
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The rank of a "triangle-free" matrix
This is a version of the question I asked recently, but the assumptions got now strengthened substantially.
Suppose that $A=(a_{ij})_{1\le i,j\le n}$ is a square matrix with all elements in $\{0,\...
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Matrix trace & norm [closed]
For any nonnegative semidefinite matrix $A$ and any matrix $B$, we have
$$\mbox{tr} (AB) \le \mbox{tr} (A) \, \|B\|$$
where $\mbox{tr}(\cdot)$ is the trace and $\|\cdot\|$ is the operator norm. How ...
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When is an algebra of commuting matrices (contained in one) generated by a single matrix?
Let C be an nxn matrix, then the polynomials in C (with appropriate coefficients) form an algebra of commuting matrices. I feel that I should know if the converse is true but I do not. So my first ...
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Vandermonde matrix is totally positive
A totally positive matrix $M\in \mathcal{M}_{n\times m}(\mathbb R)$ is such that all of its minors of all sizes are positive. It is true that any Vandermonde matrix (with well-ordered positive entries)...
- 5,428
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Are all vector-space valued functors on sets free?
Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between ...
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Necessary conditions for the existence of solution of Sylvester equation AX=XB
Let's consider square matrices $A_{n \times n}$, $B_{n \times n}$ and $X_{n \times n}$ with elements from $\mathbb{R}$. Could you tell me please, what would be the necessary conditions for the ...
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Examples of finitely presented subgroups of $\operatorname{GL}(n,\mathbb{Z})$ with unsolvable decision problems
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Does there exist a finitely presented subgroup of $\GL(n,\mathbb{Z})$ for which it is known that the conjugacy problem is unsolvable (if yes, ...
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"sinc'n determinant"
The function $\text{sinc}(x)=\frac{\sin x}x$ permeates mathematics and physics in several aspects, and it carries multiple presentations/formulations. My interest is to inject yet another one of such.
...
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Eigenvectors of a particular transition matrix
I am considering a Markov chain with $n$ states with a particularly nice structure. The transition matrix is as follows:
\begin{equation}\mathbf{P}=\begin{pmatrix}
0 & 0& \dots&0 & 0 &...
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Diagonalizing a Certain Real and Symmetric Toeplitz Matrix
Consider $0\leq \alpha\leq 1$, and let $A_{\alpha}$ be the Toeplitz $n\times n$ matrix given by
$$
A_\alpha := \begin{bmatrix}
1 & \alpha & \alpha^2 & \ldots &\alpha^{n-1} \\\
\alpha ...
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Similar matrices over $\mathbb Z_p$
Let $A$ and $B$ be two $n \times n$ matrices with entries in $\mathbb Z_p$, the $p$-adic integers. Is it true that $A$ and $B$ are conjugate iff they're conjugate over $\mathbb Q_p$ and over $\mathbb ...
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How few $k$-dimensional subspaces of $V$ are enough to have a complement to each $n-k$-dimensional subspace?
Let $n$ and $k$ be nonnegative integers such that $k\leq n$. Let $F$ be a field, and let $V$ be an $n$-dimensional $F$-vector space. A set $\mathcal{S}$ of $k$-dimensional subspaces of $V$ is said to ...
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Why does this matrix have zero determinant?
This curious identity arose from studying reductions of the maximal ideal in certain monomial algebra. It can be proved "by hand", (i.e, using Macaulay 2), but I am seeking a more conceptual ...
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Involutions in GL_n(Z)
Is there a classification of involutions in $\text{GL}_n(\mathbb{Z})$?
Here's some more details about what I mean. Consider $f \in \text{GL}_n(\mathbb{Z})$ such that $f^2=1$. Regard $f$ as an ...
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How far is a set of vectors from being orthogonal?
Given some vectors, how many dimensions do you need to add (to their span) before you can find some mutually orthogonal vectors that project down to the original ones?
Or, more formally...
Suppose $...
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How can we realize different combinatorial objects as the dimension of a construction on vector spaces? Are the resulting algebras useful?
Fix a vector space $V$ of dimension $n$ over some field $F$. Here are three commonly seen constructions:
its $k$th tensor power, $T^kV$, which has dimension $n^k$
its $k$th exterior power, $\Lambda^k(...
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Semi-linear operators
If $V_1$ and $V_2$ are finite-dimensional vector spaces over a field $E$, each equipped with an $E$-linear operator $\phi$, we can tell if $V_1$ and $V_2$ are isomorphic as $\phi$-modules by comparing ...
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Generalizing the Pfaffian: families of matrices whose determinants are perfect powers of polynomials in the entries
Let $n$ be a positive integer, and let $M = (m_{ij})$ be a skew $2n \times 2n$ matrix. That is, we have $m_{ij} = -m_{ji}$ for $1 \leq i, j \leq 2n$. Then it is well-known that
$$\det M = p(M)^2,$$
...
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A Question on Random Matrices
Consider the following $n\times n$ random matrix $V_{n}$ where the $(p,q)$ entry is given by
$$
V_{n}(p,q):= \frac{1}{\sqrt{n}}\exp(2\pi i(p-1) x_{q})
$$
where $x_{1},x_{2},\ldots,x_{n}$ are iid ...
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What is the dual concept to "annihilator" called, and do any linear algebra textbooks discuss this concept first?
When introducing dual spaces for the first time, most linear algebra textbooks proceed in what seems to me a rather backwards fashion: the annihilator $\{f\in V^*: f(u)=0\quad \forall u\in U\}$ of a ...
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Do these matrix rings have non-zero elements that are neither units nor zero divisors?
First, a disclaimer: This is a repost of a question I asked on stackexchange (no answer there).
Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ matrices with ...
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"Conjugacy rank" of two matrices over field extension
I have posted this elsewhere and got only a partial reply. I don't know whether this qualifies the question for an open-problem tag; if it does, please anyone insert it.
Let $L$ be a field, and $K$ a ...
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For a stable matrix $B$ and anti-symmetric $T$, such that $B(I+T)$ is symmetric, show that $\mbox{tr}(TB)\leq0$
Let stable matrix (i.e., its eigenvalues have negative real parts) $B \in \mathbb R^{n \times n}$ and anti-symmetric matrix $T \in \mathbb R^{n \times n}$ satisfy
$$B^\top - T B^\top = B + B T$$
...
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Lipschitz property of the determinant
$\newcommand{\A}{\mathcal A}\newcommand{\Tr}{\operatorname{tr}}$For $c$ and $C$ such that $0<c<C<\infty$, let $\A_{d;c,C}$ denote the set of all symmetric positive-definite real $d\times d$ ...
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A simple but curious determinantal inequality
Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices and $k>0$ real. Then $A^k$ is well-defined and experimentally, we have $$\det(A^k+BABA^{-1})\geqslant \det(A^k+BA^{-1}BA),$$or ...
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Finding minimum (or maximum) element of a low rank matrix.
Let $A\in\mathbb{R}^{n\times n}$ and suppose that $A$ is of rank $m\leq n$. Moreover suppose we know $u_1,\ldots, u_m \in\mathbb{R}^{n\times 1}$ and $v_1,\ldots, v_m \in\mathbb{R}^{n\times 1}$ such ...
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Is the discriminant of a free (as a module) $R$-algebra always congruent to a square modulo 4?
Let $R$ be a commutative ring. Let $A$ be an $R$-algebra (i.e., an $R$-module
equipped with an $R$-bilinear multiplication map that turns $A$ into a unital
ring). We do not require $A$ to be ...
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Perron-Frobenius theory for reducible matrices
Can someone suggest some sources/references dealing with the Perron-Frobenius theory for nonnegative matrices that are reducible?
Specifically, if $A\ge 0$ is a $d\times d$ matrix with no assumptions ...
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Number triangle
This question arose just out of curiosity. Note the triangle of 0-1's below, whose construction is as follows. Choose any number, say 53 as done here. The first line of the triangle is the binary ...
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Determinant equal to Fibonacci sequence
I need to find the determinant of matrix defined by
\begin{align*}
& a_{i,1}=a_{1,j}=1,\quad \forall 1\leq i,j\leq n,\\ & a_{i,j}=a_{i-1,j}+a_{i,j-1}+i-j, \quad \forall 1< i,j\leq n.
\...
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How flexible is the infinite-dimensional torus?
Let $\mathbb T=\mathbb R/\mathbb Z$ be the circle group and $\mathbb T^\omega$ be the infinite-dimensional torus, considered as an abelian compact topological group.
Problem 1. Is it true that for ...
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