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

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2
votes
1answer
63 views

Proof of eigenvalue stability inequality via Courant-Fischer min-max theorem

Dr. Tao in his notes on eigenvalue inequalities uses Courant-Fischer min-max theorem to prove the eigenvalue stability inequality. Specifically, I am looking for proof of Eq. (13) where Dr. Tao ...
2
votes
2answers
118 views

Is the exterior power of a primitive matrix still primitive?

the question is already in the title. Here some more details. I have a primitive matrix $M$ (primitive means $\exists k\geq 0$ such that $M^k > 0$). I take exterior powers $\wedge^n M$ and I would ...
2
votes
1answer
177 views

Lebesgue measure of set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ are linearly dependent

I've asked this question here on math.stackexchange, but I have been unable to solve this yet, so I'm hoping I can get some advice here. Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ ...
7
votes
0answers
162 views

Samuel Karlin's problem: Probability of positive solution to system of random linear equations

I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only ...
0
votes
0answers
41 views

Self adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator H acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
14
votes
1answer
786 views

How many values determine a norm?

It is well known that for a bilinear form over an n-dimensional vector space, $n^2$ values (on all pairs of basis-vectors) determine it uniquely. How many values do we need to specify in order to ...
-4
votes
0answers
42 views

Linear Algebra, Finding All Basis for B^n , etc [closed]

Let B^n be the vector space of column vectors with bit entries 0, 1. Write down every basis for B^1 and B^2. How many bases are there for B^3? B^4? Can you make a conjecture for the number of bases ...
19
votes
3answers
951 views

Dividing by two in the category of vector spaces

Does every invertible linear map $M$ between $V \oplus V$ and $W \oplus W$ naturally yield an invertible linear map $L$ between $V$ and $W$? Here "naturally" means "in an $GL(V) \times ...
0
votes
2answers
62 views

Symmetric matrix from a nonsymmetricc matrix

Basically this is a part of a long algorithm to calculate some matrix properties. Given an upper triangular square matrix R, how can I find an orthonormal matrix W (possibly iteratively) such that WR ...
1
vote
0answers
62 views

Bounding the largest Singular value

D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$. B is any $n \times n$ n.n.d. matrix. What will be the sharpest upper bound on the largest eigenvalue of: ...
2
votes
0answers
38 views

Smallest Singular Value of a Random Matrix with Dependent Entries

Overview I am trying to bound from below the smallest singular value $\sigma_{n}$ of a sequence of symmetric $n$ by $n$ random matrices $M_{n}$ with dependant entries. In particular, I would like to ...
1
vote
0answers
45 views

Complex Hessian Signature

It' all, simply, about the signature of a matrix. Let $\Omega\subseteq\Bbb C^n$ open, $r:\Omega\to\Bbb R$ twice differentiable (real differentiable, not necessarely complex differentiable, i.e. not ...
2
votes
1answer
105 views

Find a line such that sum of perpendicular distances of points to the line is minimized

Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
-1
votes
0answers
14 views

Is a complex vector space closed under complex conjugation? [migrated]

Given a complex vector space $\mathcal{V}$, its complex conjugate $\overline{\mathcal{V}} = \{ \overline{v} : v \in \mathcal{V} \}$ consists of the "same" set of points (according to a number of ...
1
vote
0answers
62 views

Minimize Product of Sums of Squared Distances

The Question Given two sets of vectors $S_1$ and $S_2$,we want to find a unit vector $s$ such that $$\{\sum_{u\in S_1}(\|u\|^2-\langle u, s \rangle^2)\} \cdot \{\sum_{v\in S_2}(\|v\|^2 - \langle v, ...
0
votes
0answers
59 views

About expectation norms on graphs

Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= ...
1
vote
2answers
108 views

Solving $Ax=e_k$ for standard basis vector $e_k$, sparse $A$

Given a sparse matrix $A \in \mathbb{R}^{n \times m}$, are there any efficient methods for determining whether there exists an $x \in \mathbb{R}^m$ such that $Ax=e_k$, the $k^{th}$ standard basis ...
1
vote
0answers
24 views

Volume under the intersection of scaled simplices

This is rather specific but I need to compute the volume under the intersection of rescaled simplexes, that is, the volume of the space: $\left\{x \in \mathbb{R}^n|\sum_i c_{ki} |x_i| \leq1\; k = 1 ...
0
votes
1answer
103 views

Minimal dimension of a Lie algebra of matrices, with a restrictive property

Let $\mathfrak{g}$ be a sub-Lie-algebra of $\mathfrak{gl}_n(\mathbb{C})$, the Lie algebra of complex $n\times n$ square matrices. Let us call $(H)$ the hypothesis: for all $x, y\in\mathbb{C}^n$, ...
0
votes
0answers
7 views

dual basis of cohomology algebra [migrated]

Let $H^*(M)$ be the cohomology algebra of oriented manifold $M$ with rational coefficients. Let $\{b_i\}$ be a basis of $H^*(M)$ as a vector space over $\mathbb{Q}$. Let the dual basis be ...
1
vote
0answers
68 views

Notions of consistency / heterogeneity in sets of vector values?

The problem Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$ I ...
1
vote
1answer
99 views

Dense symmetric unitary integer matrix?

Can someone give me a nontrivial example of a symmetric unitary integer matrix? I'm looking for something as dense as possible (i.e., not too many 0's); 5 <= size <= 8 would be ideal.
0
votes
0answers
72 views

Hadamard / matrix product adjoint

First of all I would like to thank everyone over here at mathoverflow for their amazing generosity and help (for both pros and newbies like myself). I apologize if this question seems dumb; I'm a new ...
1
vote
0answers
71 views

Extension of scalars and projective limits

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor ...
6
votes
5answers
292 views

The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$. Is there any ...
1
vote
0answers
87 views

Showing a wedge product is nonzero

Let $V$ be a complex vector space of dimension $n$, equipped with a Hermitian inner product whose Kahler form we denote by $\omega$. Let's set $P = \bigwedge^{2p} V^*$ and $Q = \bigwedge^{2q} V^*$ for ...
1
vote
1answer
73 views

Expectation of Gaussian random vector & arbitrary function thereof?

I saw in a paper (https://www.princeton.edu/~wbialek/rome/refs/bialek+ruyter_05.pdf Eq.37) the following identity: where the <.> operator refers to a population average. No source or ...
7
votes
0answers
264 views

Does this inequality always hold?

Denote the adjacency matrix of a given undirected graph by $g$. It is an $n$-by-$n$ symmetric Boolean matrix with elements on the diagonal to be zero ($n\geq 3$). Let $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...
4
votes
0answers
73 views

Geometric interpretation for partial trace?

This MO question asks for a geometric interpretation of the trace of a linear transformation. I'm wondering about a geometric interpretation of partial trace. Given a linear transformation $f: ...
3
votes
1answer
167 views

Matrix Elements of Real Representations

I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer. Suppose that $G$ is a finite group and we have a unitary irreducible representation ...
0
votes
0answers
97 views

About distinct eigenvalues of a graph

if a graph with adjacency matrix $A$ and Laplacian $L$ has $k$ distinct eigenvalues then does this fact naturally help define or prove existence of a polynomial $p$ of degree $k-1$ such that ...
2
votes
1answer
47 views

Reducing eigenvalues of symmetric PSD matrix towards 0: effect on ratios of original matrix elements?

Let $\boldsymbol{S}$ be $k \times k$ positive semi-definite real symmetric matrix with eigen decomposition $\boldsymbol{S} = \boldsymbol{X} \boldsymbol{\Lambda} \boldsymbol{X}'$ ...
5
votes
1answer
157 views

Dimensions of a vector space akin to modular symbols

The group $\operatorname{SL}_2(\mathbb Z)$ acts on polynomials in two variables $\mathbb C[x,y]$ via $A\cdot f(x,y)\mapsto f(A^{-1}.(x,y))$ where $(x,y)$ is regarded as a column vector. There are two ...
0
votes
0answers
34 views

Changes in singular Values of matrix when adding row

I know that if a column is added to a matrix then the matrix largest signular value increases and the smallest singular value decreases. That is: Given matrix $A \in R^{m \text{x} n}$, $m>n$, and ...
0
votes
0answers
21 views

Determinant of special diagonal matrix [migrated]

Suppose A is a square matrix of any order whose diagonal entries are integers which are pairwise prime and all other entries are 1. Is there any method to find determinant of this matrix A.
3
votes
0answers
78 views

Bound on the ratio of top 2 eigenvalues

Let $P$ be a $(n+1) \times (n+1)$ stochastic matrix such that $P_{ij}=\tau$ if $i \neq j$ and $P_{ii} = (1 - n\tau)$ where $0<\tau < \frac{1}{n+1}$. It is clear that the largest eigenvalue of ...
2
votes
2answers
125 views

Matrix inequality

Let $\mathbf{Z,R}$ two Hermitian semidefinite positive matrices with all eigenvalues larger than one. Intuition drives me that $\mathbf{R}^{-1/2}\mathbf{Z} \left(\mathbf{R}^{-1/2}\right)^H - ...
5
votes
0answers
86 views

A weak Perron-Frobenius property for sets of positive matrices

A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
0
votes
0answers
14 views

On the stability analysis of a discrete difference system with multiplicative noise

If we assume that \begin{equation*} \rho \{\phi \otimes \phi+\psi \otimes \psi\}<1 \end{equation*} where $\rho$ denotes the spectral radius, then can we verify the following inequality holds ...
1
vote
1answer
67 views

Computation of extreme rays of rational polyhedral cones - Hemmecke's project and lift algorithm

I am working on an implementation of Raymond Hemmecke's algorithm for finding generating sets of cones: http://arxiv.org/abs/math/0203105 Unfortunately I am struggling to make the algorithm work on ...
0
votes
1answer
195 views

Suppose that $G$ is a subgroup of $GL_n(\mathbb C)$ with finite exponent. Then is $G$ a finite group? [closed]

As title. the exponent of $G$ is the least number $n$ (if exists) such that $g^n=e$ holds for all $g\in G$ or $+\infty$.
2
votes
2answers
135 views

What is the maximal number of sub spaces of a fixed dimension such that there is another sub space which intersects them are all null

Let $\mathbb F_q$ be the finite field with $q$ elements. Suppose $V$ is a linear space of dimension $n$ over $\mathbb F_q$, and $r<n$. What is the maximal $k$ such that for arbitrary $k$ subspaces ...
1
vote
0answers
146 views

Is there a method to simultaneously block-diagonalize a set of group matrices?

Assume that you are explicitly given the representation matrices of a group. How does one go about finding that common basis which will find the irreducible components of all of them simultaneously? ...
1
vote
0answers
51 views

What are good bounds on ratios of subdeterminants?

Let $A$ be a symmetric matrix and $A_i$ be the matrix obtained from $A$ by dropping the $i^{th}$ row and column. Then what are some good bounds on the value of $\frac{det(A_i)}{det(A)}$ ? Using the ...
5
votes
1answer
152 views

The Maximal $\ell_2$ norm of a signed sum of vectors

Suppose we have $n$ vectors in $\mathbb{R}^n.$ Consider the signed sum of these vectors: $$U(s_1,\ldots,s_n)=s_1 v_1+s_2 v_2 + \ldots + s_n v_n$$ where $s_j$'s can only take values of $+1$ or $-1.$ I ...
0
votes
1answer
72 views

Scalar restriction and scalar extension

Consider a morphism of commutative rings $h\colon R\rightarrow S$. This yields the two functors $h_*\colon{\sf Mod}(S)\rightarrow{\sf Mod}(R)$ (scalar restriction) and $h^*\colon{\sf ...
0
votes
0answers
57 views

About bounding values of quadratic forms

It would be helpful if someone can share (either as references) examples of calculations/analysis which achieves bounding of values of quadratic forms in say either of the following situations, ...
1
vote
1answer
65 views

Traces and projectors

Suppose that $V$ is a $\mathbb C$-vector space. I'm eventually interested in the infinite-dimensional case, but let's say for now that it's finite dimensional. Suppose that $\mathscr S$ is a ...
4
votes
2answers
200 views

Fixed space of the square of a symmetric matrix over $\mathbb{F}_2$

Let $M$ be an invertible symmetric $2n \times 2n$ matrix with entries in the finite field $\mathbb{F}_2$. Is $\mathrm{Ker}\ (M^2 - I_{2n})$ necessarily even dimensional?
1
vote
1answer
41 views

proving that a smooth curve in Euclidean n-space contains n+1 affinely independent points

If I let $f(\theta)=((\mathrm{cos} \theta)X+(\mathrm{sin} \theta)Y)^{n-1}$ and view the range of this curve as a subset of the space of homogeneous polynomials of degree $n-1$ in two variables viewed ...