0
votes
1answer
35 views

Find the number of connected components of the set of real $2\times 2$ matrices of determinant $1$ which keep $x^2 - y^2$ invariant

Working on yet another past comprehensive exam question. Let $S$ be the set of real $2\times 2$ matrices with determinant $1$, keeping invariant the form $x^2 - y^2$. Regard $S$ as a subset of ...
2
votes
2answers
127 views

invertible matrices connected or not

The question asks "Is the set of all 3 by 3 real invertible matrices connected or not?" My intuitive idea is that we can establish a separation consisting of matrices with positive and negative ...
1
vote
2answers
30 views

Set of real symmetric matrices with signature $(2,1)$ is open

Let $S$ be the space of all $3\times 3$ real symmetric matrices, let $B$ be the subset of $S$ with signature $(2,1,0)$. Show that $B$ is open in $S$ in the topology of $\mathbb{R}^6$. My thoughts: ...
0
votes
0answers
61 views

Zero set of finitely many polynomials.

Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed ...
4
votes
1answer
46 views

Closure and compactness of the set of real eigenvalues ​​of a real matrix.

Let $A$ be a part of $\mathcal{M}_n(\Bbb{R})$ and $B$ the set of real eigenvalues ​​of the matrix $A$. 1) Show that if $A$ is compact then $B$ is compact as well. 2) If $A$ is closed ...
1
vote
0answers
27 views

Conjunctive Normal Form representation/ First Order Logic.

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: " y Cooperates with x" relationship: means if there is two ...
0
votes
1answer
44 views

Are circulant matrices open

Are the set of positive definite symmetric circulant matrices open in the set of positive definite symmetric matrices?
0
votes
0answers
30 views

Find close points by grouping points in n-dimensional space

I know I already asked a similar question over here: Finding the closest point in a set to another point in n-dimensional space: efficiently But now my question is different. I have many points ...
1
vote
2answers
46 views

Some questions about the proof of the General Linear Group being a manifold.

I understand the idea behind proving that GL(n,$\mathbb{R}$) is a smooth manifold by first using the fact that it is isomorphic to $\mathbb{R}^{n^{2}}$ and using the continuity of the determinant ...
0
votes
1answer
28 views

If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable.

If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable. It seems very obvious intuitive, but how to write a good solid proof? Notice I take the closure of the span (the ...
1
vote
1answer
51 views

Finding the closest point in a set to another point in n-dimensional space: efficiently

I'm a programmer and am working on writing an efficient algorithm that, given a point P in n-dimensional space, can find the closest point from a set of points. For ...
5
votes
0answers
102 views

Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
1
vote
1answer
41 views

Finite family of subtori in the torus $(S^{1})^{n}$

Working on a problem on matroids, I've already ask a question about some subtori. Here's the link to a previous problem: Topological subspace in $(S^{1})^{n}$ Anyway, here's another problem related ...
1
vote
1answer
49 views

Topological subspace in $(S^{1})^{n}$

Studying the set of solutions of a particular linear system associated to a matroid, I notice that is it possibile to determine the topology of the quotient and identify it as a subtorus of ...
1
vote
1answer
61 views

Elementary proof that $Gl_n(\mathbb R)$ and $Gl_m(R)$ are homeomorphic iff $n=m$ [duplicate]

Prove that $Gl_n(\mathbb R)$ and $Gl_m(R)$ are homeomorphic iff $n=m$ Since $Gl_n(\mathbb R)$ is homeomorphic to an open subset of $\mathbb R^{n^2}$, this boils down to proving that two open ...
3
votes
1answer
44 views

Convex hull of $\exp\bigl( \mathcal{M}_n(\mathbb R)\bigr)$

What is the convex hull of $\exp\bigl( \mathcal{M}_n(\mathbb R)\bigr)$ ? My attempt : lemma For $A\in\mathcal{M}_n(\mathbb C)$ there exist $P(X)\in \mathbb{C}[X]$ such that $A=\exp{P(A)}$ ...
0
votes
0answers
31 views

A peculiar fact about 3-dimensional complex projective space

I'm working on a result for my master's thesis, that right now involves translating a proof I don't quite follow, to something that is a bit more in line with what I already know. We define ...
1
vote
1answer
29 views

Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
-1
votes
3answers
79 views

Let $W$ be a linear subspace of dimension $n-1$ in $\mathbb R^n$. Is $W$ closed in $\mathbb R^n$? [closed]

Let $W$ be a linear subspace of dimension $n-1$ in $\mathbb R^n$. Is $W$ closed in $\mathbb R^n$? Also, is it open?
10
votes
2answers
129 views

Let $A$ and $B$ in $O_n(\mathbb{R})$. Show that $A$ and $B$ commute.

Let $A$ and $B$ in $O_n(\mathbb{R})$ (orthogonal matrices) such that $|||B-I_n|||<\sqrt{2}$ (subordinate norm) and $A$ commute with $BAB^{-1}$. Show that $A$ and $B$ commute. My ...
0
votes
3answers
30 views

Some questions of vectors and dense subsets

I have a couple of quick functional analysis related questions: 1.Say we have a normed space $V$ and reflexive, separable Banach space and $K \subset V$ a closed, convex, bounded subset of $V$. ...
-1
votes
1answer
55 views

chosing between matrix theory and combinatroics

I have to take one more math course to finish my math minor , i am a computer science major and i want to know which course will benefit me more matrix theory or combinatorics and which takes more ...
1
vote
1answer
61 views

Maps from $SO(3)$ to $S^1, S^2$, and $S^1 \times S^2$

I am looking for continuous maps between the special orthogonal group of $3\times 3$ matrices and the unit circle, unit sphere, and their product ($S^1$, $S^2$, $S^1 \times S^2$, respectively). Any ...
0
votes
2answers
67 views

Dense Countable basis on Hilbert space

Let say that I have a $H$ hilbert space and linear independent countable set $\beta =\{ \beta_1 , \beta_2, \beta_3... \}$ such that $span(\beta)$ is dense set in H. does $span(\beta-\beta_1) =span( ...
1
vote
1answer
51 views

Computing the intersections of arbitrary number of planes

Imagine three sets of planes (A, B, and C). The planes in each set are parallel and evenly spaced and so can be defined by a plane equation and a scalar representing the distance between planes. If ...
0
votes
0answers
15 views

Constructing a smoothly varying basis without singularities

I am trying to construct a smoothly varying and a differentiable basis to map a vector in $\mathbf{B}:\mathbb{R}^3 \to \mathbb{R}^3$. Given a vector field $\mathbf{n}(\mathbf{x})$ where $\mathbf{n} = ...
1
vote
1answer
40 views

the “unit speed” anlogue of the evolute of the curve

Given a curve, $\gamma: \mathbb{R} \to \mathbb{R}^2$ define the flow in the normal direction by $\gamma(t) + \epsilon \, \mathbf{n}(t)$. This is different from the evolute which moves at speed ...
2
votes
2answers
46 views

Using a contradiction to show something is not compact

Consider the set of $3$ by $3$ matrices in $\mathbb{R}$ that have nonzero determinant. I want to prove that this is not compact preferably with a contradiction. Attempt: Since its a determinant it ...
0
votes
1answer
32 views

Relations between an affine space and a topological space

What is the relation between an affine space and a topological space? Is one a specialization of the other? Moreover, what do we call a point in geometry: an element of a topological space or an ...
-1
votes
1answer
26 views

Finding a condition to make a map a contracting map

Consider $(\mathbb{C}^n ,||\cdot||_{1})$ where $||x||_{1} = \sum\limits_{i=1}^{n} |x_{i}|$ and $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
0
votes
1answer
57 views

Is the set of all matrices in M(n;R) all of whose eigenvalues satisfy the condition |λ|≤2. compact?

Is the set of all matrices in $M(n; R)$ all of whose eigenvalues satisfy the condition $|λ| ≤ 2.$ compact?
0
votes
1answer
47 views

Hypersphere isometry?

I will denote the $n-$sphere of radius $1$ centered at the origin as $\mathbb{S}^n$, so that $$ \mathbb{S}^n = \{ x \in \mathbb{R}^{n+1}\ : \ \|x\| = 1\}. $$ I am stuck on the following problem...I'm ...
2
votes
3answers
117 views

Positive definite part of a symmetric matrix - or: are the positive definite matrices a retract of the set of symmetric matrices?

$\newcommand{\Sym}{\operatorname{Sym}}$ Denote by $\Sym(n)$ the set of symmetric, real $n\times n$ matrices and let $\iota:\Sym^+(n)\hookrightarrow \Sym(n)$ be the subset of positive definite ...
2
votes
3answers
96 views

topology on $\hom_{\mathbb C}(V,W)$

Let's $V,W$ be finite-dimensional complex vector spaces. How to define topology on $\hom_{\mathbb C}(V,W)$? As I see we can define metric in this $(\dim_{\mathbb C}V\times\dim_{\mathbb ...
1
vote
1answer
30 views

Construct dense and disjoint sets of $\mathbb{R}^m$ so that every element of their Cartesian product has full rank

Or equivalently, can one construct sets $S_1 ,S_2 ,\dots ,S_n \subseteq \mathbb{R}^m$ so that (i) the sets $S_i$ are dense and disjoint; and (ii) if one picks from each set $S_i$ any element $u_i$, ...
1
vote
2answers
56 views

$SO(3)$ with minimal and maximal trace.

Let $O(3)$ be the set of $3 \times 3$ orthogonal matrices. Let $SO(3)$ be a subset of $O(3)$ such that det($A$)=1 for all $A \in SO(3)$. Show that there is a matrix with minimal trace in $SO(3)$ and ...
2
votes
1answer
67 views

Infinite line is closed in $\mathbb{R}^n$

I have been reading the book "Elements of the functional analisys", by Kolmogorov and Fomin. At the chapter of Normed Linear Spaces, page 73 to be precise, the author makes the following definitions: ...
2
votes
2answers
138 views

Questions on positive definite matrices

First, in this discussion, I am only considering real matrices. Second, I have a few questions I am ruminating on related to symmetric matrices. Some of these questions I need someone to say my ...
3
votes
1answer
67 views

Giving an explicit example of a vector that is perpendicular to $v$

Let $v\in\mathbb{R}^3$ be a unit vector. It is possible to show that there exists vectors $\{w_1,w_2\}$ such that $\{v,w_1,w_2\}$ is orthonormal by applying the Gram-Schmidt process, but can we do so ...
1
vote
0answers
16 views

Quotient space of $End(\mathbb{R}^n)$

Let $X=End(\mathbb{R}^n)$ be the set of linear maps $\mathbb{R}^n\rightarrow\mathbb{R}^n$. We endow $X$ with a topology, induced from the metric on $X=\mathbb{R}^{n^2}$. I want to understand the ...
0
votes
1answer
423 views

Convexity of sum and intersection of convex sets

Let $A_i$ be a subset of $\Bbb{R}^m$ which is convex for $i=1,...,n$. How can I prove that the sum of $A_i$ is also convex? I know how to prove it with two sets: Let $x = a_1 + b_1$ and $y = a_2 ...
1
vote
1answer
63 views

For $k \times n$ matrices $X,Y$ of rank $k$, find a characterization of $Y = AX$, $A \in \operatorname{Gl}(k, \mathbb{R})$

Let $F(k,n)$ (with $k < n$) denote the set of $k \times n$ matrices of of rank $k$ (entries in $\mathbb{R}$). We can give $F(k,n)$ a Euclidean topology by identifying it with a subset of ...
3
votes
1answer
421 views

Is this reading path recommended?

Since doing math requires learning it first, I 've chosen a series of books to understand some ''Higher math''(which I want to read over a period of several years),and would like to see some ...
7
votes
2answers
132 views

Is a homeomorphism which maps lines to lines (and fixes zero) necessarily linear?

We know the homeomorphism $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ maps straight lines to straight lines and the zero vector to the zero vector. Is it Linear?? If so, how can we prove it?
0
votes
0answers
62 views

Linear map preserves closedness of a convex set with property on recession cone

Let $\mathbf{E},\mathbf{Y}$ be two euclidean space, $C$ be a non-empty closed convex set in $\mathbf{E}$. The map $A:\mathbf{E}\rightarrow\mathbf{Y}$ is linear, and $N(A)\cap 0^+(C)$ is a linear ...
1
vote
1answer
78 views

Help with a proof of a theorem about convex sets

I'm studying the following theorem: Theorem 1. Let $C$ be a convex set and let $\textbf{y}$ be a point exterior to the closure of $C$. Then there is a vector $\textbf{a}$ such that ...
2
votes
2answers
322 views

Prove that invertible metrices set is an open set in a given space, and the determinant is continuous [duplicate]

Given a matrix $M_{n\times m}$, we can think about it as a vector in $\mathbb{R}^{n\times m}$ (How come?). How can I prove that the set of all the invertible metrices of size $n\times n$ is an open ...
0
votes
1answer
84 views

Set of all matrix of rank $ r $ is open set in $ M_n (\mathbb { R })$

I have no idea how to start it. Actually I have no idea which matrix in $ M_n (\mathbb {R})$ are of rank $ r $. I know all basic result about it. please help me.
6
votes
5answers
226 views

This set of matrices is open

I'm trying to prove that the set of the matrices whose eigenvalues have non-zero real part is an open subset of $M^n$, the set of square matrices with order $n$ which is identify with $\mathbb ...
0
votes
0answers
172 views

What does is mean by differentiate a matrix $E - DB^{-1}C$?

The problem I am trying to solve is: Prove that the set of $m \times n$ is matrices of rank $r$ is a submanifold of $\mathbb{R}^{mn}$ of of codimension $(m - r)(n -r)$. [HINT: Suppose, for ...