# Tagged Questions

**0**

votes

**1**answer

24 views

### Linear independence of +/- 1 strings/vectors

Let $V=\left\{-1,1\right\}^{n}$. Consider three vectors $v_1,v_2,v_3\in V$. I would like to know whether these vectors are linearly independent over $\mathbb{Z}$. To be more precise - I need a ...

**4**

votes

**3**answers

275 views

### Analogue of Cayley Hamilton theorem for operators on Hilbert space

Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.

**5**

votes

**1**answer

156 views

### Dimension of the span of all partial derivatives of a given symmetric polynomial $f$ and the polynomial $E(f)$

I need some help on the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ ...

**3**

votes

**0**answers

138 views

### How to prove the following determinant identity?

This problem is relevant to the spin operator matrix elements in the quantum 1D XY model.
For any even integer $N$, define two sets ...

**0**

votes

**0**answers

88 views

### Can we assume Power spectrum of a signal as element/member of Polish space? [on hold]

Absolute value of short time Fourier transform gives the power spectrum of a signal compact on time. It cannot distinguish between noise and signal, further we also get the side-lobe. The side lobe ...

**-5**

votes

**0**answers

93 views

### What is a good book for reviewing high school math, and preparing for university? [on hold]

I'm signing up for University soon (Compsci program) as a mature student. It's been a long time since I've done any math, and I went as far as grade 11 in high school. So, I'm looking for a book that ...

**2**

votes

**3**answers

321 views

### A table for irreducible integral representation of finite cyclic groups

Is there such a table where the irreducible integral representations of finite cyclic groups
are listed?
Edited:
Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...

**1**

vote

**1**answer

86 views

### Projectors onto the invariant subspaces of a unitary representation $U \otimes U^* \otimes U \otimes U^*$

Let $$U \mapsto U \otimes U^* \otimes U \otimes U^*$$ be a unitary representation of the unitary group $U(n)$ acting on the vector space $V$ (where $U^*$ is the complex conjugate of $U$). We can ...

**-1**

votes

**0**answers

85 views

### Show: Real roots of a polynomial [on hold]

I´ve got some problems with this task I found in a german script.
Be $\ N \in \mathbb{N}$. Define matrix $\ \Pi := (\pi)_{0\leq i,j \leq N}$ with
\begin{equation}
\pi_{i,j} := \binom{N}{j} p_i^j ...

**37**

votes

**7**answers

2k views

### How to prove this determinant is positive?

Given the matrices $ A_i=
\biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)
$, where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove that $\det(I + ...

**1**

vote

**0**answers

30 views

### Is there a relation between covariance matrices and real and imaginary part of eigenvectors?

Apology if my question not clear or appropriate. Consider a complex positive definite sample covariance matrix (SCM) A, generated by a band limited signal on a set of sensors which is termed as data ...

**1**

vote

**1**answer

33 views

### How to characterize a linearly-constrained subspace in a projection [closed]

I hope this one is easy. Suppose I have an underdetermined, rectangular matrix $A$ and vector $b$. I want to reason about the subspace where $Ax = b$ and specifically the projection $y:= Tx$. Is there ...

**4**

votes

**0**answers

117 views

+50

### What is the complexity of intersecting two matrix algebras over a finite field?

The following question arose in a joint project with Arkadius Kalka and Adi Ben-Zvi.
Let $\mathbb{F}$ be a finite field, and $M_n(\mathbb{F})$ be the $n\times n$ matrices over $\mathbb{F}$.
For a ...

**2**

votes

**2**answers

128 views

### constant rank theorem for banach spaces

Is there a similar statement to the constant rank theorem for finite dim real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dim Banach space ...

**0**

votes

**0**answers

41 views

+50

### Convenient Basis Presentation of Lefschetz Decomposition

Let $V$ be an almost-complex vector space, equipped with a symplectic element $\omega \in V^{(1,1)}$. In terms of a basis $b^+_i \in V^{(1,0}$, $b^-_i \in V^{(0,1}$, does there exist a "simple" ...

**5**

votes

**1**answer

114 views

### Number of matrices with given Smith normal form

Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...

**0**

votes

**0**answers

33 views

### Representing a complex graph with a single formula [closed]

We have specific rules for calculating mark-up on our products. For one product line, we have a 15% mark-up until the margin reaches \$50, at which point we keep the margin at /$50 regardless of how ...

**-5**

votes

**0**answers

39 views

### How do I show that {R}^{nxn} = {R}_{sym}^{nxn} + {R}_{skew}^{nxn} [closed]

How can I show that $\mathbb{R}^{nxn} = \mathbb{R}_{sym}^{nxn} + \mathbb{R}_{skew}^{nxn}$, where $\mathbf{} \mathbb{R}_{sym}^{nxn} = \{ A \in \mathbf{R}^{nxn} | A^{t} = A\}$ and $\mathbf{} ...

**-1**

votes

**0**answers

12 views

### Is it possible to create hierarchy basis? [migrated]

An eigenbasis is defined as basis consisting entirely of eigenvectors of a linear transformation. On the other hand a Schauder basis is also a basis except they allow for infinite sums. I could not ...

**2**

votes

**0**answers

75 views

### Do copairings provide dualities in derived categories?

Here is an elementary fact about vector spaces. Let $V,W$ be vector spaces over a field $\mathbb K$ and let $c : \mathbb K \to V \otimes W$ be an element of the tensor product. Then $c$ determines ...

**-1**

votes

**0**answers

28 views

### Improvement of Minimum description length (MDL) estimate [closed]

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response.
Let me consider ...

**-4**

votes

**0**answers

65 views

### Is there Practical/Physical problem solved in Banach space? [closed]

Is there a practical/physical problem that can be solved using Banach space which could not be solved using Hilbert space? This is equivalent is there a practical/physical problem which cannot be ...

**1**

vote

**0**answers

28 views

### Lattice-isotopic essentialization of arrangements

I'm working on a problem related to
$\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...

**0**

votes

**0**answers

102 views

### Symmetric kernel of tensor product

Let $V,W$ be two vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with disjoint kernels $K_i$ of dimension $1$.
Consider the tensor product of these maps $L_1\otimes ...

**2**

votes

**0**answers

34 views

### A tensor equation related to an invariant of a diffeomorphism

Let $M$ be an $n$-dimensional differentiable manifold, $f : U
\rightarrow V$ a diffeomorphism between open neighbourhoods $U$, $V$
of $M$ with $f(x)=x$ for some $x \in U$, and let $R$, $S$, $T$ be
...

**1**

vote

**0**answers

21 views

### LU growth factor applied to LDL of a Positive Semidefinite matrix [closed]

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...

**1**

vote

**0**answers

77 views

### When can the rank of a submodule be bigger than the rank of the module itself? [migrated]

It is well known that the dimension of a subspace is less than or equal to the dimension of the vector space it is contained in. The same is true e.g. for modules over a principal ring.
I am looking ...

**3**

votes

**1**answer

89 views

### Create matrix containing values in [0,1] where sum of all diagonals and anti-diagonals is fixed

The problem I am facing sounds at first glance pretty simple. However, as very often, it seems more complicated than I first assumed:
I want to calculate a matrix $P = (p_{j,k}) \in \mathbb{R}^{n ...

**7**

votes

**1**answer

144 views

### Determinant of a checkerboard Hankel matrix with Catalan numbers

My goal is to compute
\begin{equation}
I = \det \left(\mathbf{I} + \mathbf{A}\right)
\end{equation}
where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers:
$$
\left\{
...

**1**

vote

**0**answers

65 views

### Is my particular finite dimension Toeplitz matrix always strictly positive?

Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$.
Define a sequence of banded ...

**2**

votes

**1**answer

79 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

**2**answers

125 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

**1**answer

216 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 $n\times n$ ...

**8**

votes

**0**answers

198 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 ...

**1**

vote

**1**answer

59 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

**1**answer

822 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 ...

**22**

votes

**4**answers

1k 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

**2**answers

63 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

**0**answers

72 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

**0**answers

45 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

**0**answers

52 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 ...

**3**

votes

**1**answer

140 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

**0**answers

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

**0**answers

67 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, ...

**1**

vote

**1**answer

168 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

**2**answers

109 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

**0**answers

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

**1**answer

104 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 $x, y\in\mathbb{C}^n$, ...

**1**

vote

**0**answers

72 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

**1**answer

100 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.