**7**

votes

**4**answers

3k views

### 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 ...

**0**

votes

**0**answers

25 views

### limit of matrix inverse (related to an MMSE matrix)

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 ...

**-3**

votes

**0**answers

38 views

### Boundary Value System.

The boundary value problem: $$y'' + Q(t)y = f(t)$$
satisfying $$Ay(a) +By(b) = g$$
where A, B and Q are the matrices of order n.
After calculation, we can get the form of solution will be
$$y(x) = ...

**2**

votes

**1**answer

329 views

### Diagonalization of 4th order tensors

I have been wondering about the following problem...
Let $n$ be a positive integer and denote by $M_n^s$ the space of symmetric $n\times n$ real matrices. Now, we look at the space $\mathcal ...

**21**

votes

**4**answers

23k views

### Eigenvalues of Matrix Sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum?
What about the special case when they are Hermitian and positive-definite?
I am investigating ...

**14**

votes

**1**answer

793 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 ...

**2**

votes

**1**answer

187 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 $n\times n$ ...

**1**

vote

**1**answer

44 views

### Phase of the inner product between the elements of an ETF

I am doing research in compressive sampling for Cognitive Radio applications. While working on a project I came across with the following question:
Is there any research about the phase of inner ...

**2**

votes

**2**answers

120 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

65 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 ...

**1**

vote

**1**answer

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 ...

**7**

votes

**0**answers

167 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

**0**answers

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 ...

**-4**

votes

**0**answers

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

**3**answers

956 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 ...

**1**

vote

**1**answer

158 views

### Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$.
The goal is to return $\widehat A, \widehat B$ such ...

**1**

vote

**1**answer

269 views

### On permutation of elements of two bases of a vector space (Greub´s book)

Let {a1,a2,...,an} and {b1,b2,...,bn} be two bases for a vector space E. Fix p, 1 ≤ p ≤n. Is there a permutation σ such that
{a1,a2,...,ap,bσ(p+1),...,bσ(n)} and ...

**0**

votes

**2**answers

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

**1**answer

120 views

### The norm of a Finite Hilbert matrix

Let $H$ be an $n\times n$ Hilbert matrix,
$$h_{ij}=(i+j-1)^{-1}.$$
The matrix $p$-norm corresponding to the p-norm for vectors is:
$\left \| A \right \| _p = \sup \limits _{x \ne 0} \frac{\left ...

**4**

votes

**2**answers

306 views

### Eigenvectors of a symmetric positive definite Toeplitz matrix

I wish to efficiently compute the eigenvectors of an n x n symmetric positive definite Toeplitz matrix K. A full eigendecomposition would be even better.
Although I assumed this would be a well ...

**1**

vote

**0**answers

63 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:
...

**1**

vote

**1**answer

808 views

### Conjugate Matrix

Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix ...

**2**

votes

**0**answers

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 ...

**2**

votes

**1**answer

108 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**

vote

**0**answers

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 ...

**7**

votes

**0**answers

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 $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...

**2**

votes

**1**answer

123 views

### Rank changes with matrix edits

Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric).
Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix.
Case $1$: $M+W\in\{0,1\}^{n\times n}$.
Could ...

**-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

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, ...

**1**

vote

**2**answers

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 ...

**0**

votes

**0**answers

61 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)= ...

**5**

votes

**1**answer

236 views

### A specific Vandermond matrix [closed]

Consider the Vandermond matrix
$$
V (x_1, x_2, \ldots , x_n) =
\begin{pmatrix}
1 & x_1 & x_1^2 & \cdots & x_1^{n-1} & x_1^n & x_1^{n+1} & \cdots \\
1 ...

**1**

vote

**5**answers

792 views

### Presentation of the Clifford group by generators and relations?

The Clifford group $\mathcal{C}_n$ is a matrix group on $\mathbb{C}^{2^n}$ generated by tensor products of the following matrices:
$$
P = \begin{pmatrix} 1 & 0 \\\\ 0 & i\end{pmatrix}
\quad
H ...

**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 ...

**9**

votes

**2**answers

966 views

### An algorithm to find non-trivial linear dependencies

This question is inspired by another MO question about special stratifications of equivariant Grassmannians, that turned out to be a problem of computing non-trivial circuits in a vector matroid. To ...

**0**

votes

**1**answer

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

**1**

vote

**0**answers

69 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 ...

**0**

votes

**0**answers

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
...

**0**

votes

**1**answer

279 views

### Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here.
I'm ...

**2**

votes

**2**answers

4k views

### Submultiplicative matrix norm: Max Norm

Various sources claim that a maximum norm $||A||_{max}=\max_{i,j}|a_{ij}|$ is not submultiplicative, i.e. $||AB||_{max}\not\leq||A||_{max}||B||_{max}$.
Where can I find what norm a,b satisfy ...

**1**

vote

**1**answer

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.

**9**

votes

**1**answer

347 views

### Why does this antisymmetric product factor out a determinant?

Consider a generic $n \times n$ matrix $M$.
Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$:
$$R_q = M_q^T (M_q ...

**4**

votes

**3**answers

533 views

### Algorithm for the intersection of a vector subspace with a cone of non-negative vectors

Hi,
I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...

**0**

votes

**0**answers

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 ...

**3**

votes

**1**answer

256 views

### How to calculate the square root of matrix $A+B$ perturbatively?

$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$
Note that the perturbative calculation of square root ...

**11**

votes

**3**answers

1k views

### 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 ...

**0**

votes

**1**answer

205 views

### Comparing iterative methods for linear systems

For a tridiagonal matrix of the from
\begin{bmatrix}
a & -b & \newline
-b & a & -b \newline
& \ddots & \ddots & \ddots \newline
& & & & ...

**3**

votes

**2**answers

312 views

### Distributing the Hodge map over the wedge product

Let $(V,\langle,\rangle)$ be a finite dimensional inner product space, $V^{\wedge}$ it exterior algebra, and $\ast$ the Hodge star arising from $\langle,\rangle$. Does there exist any formula to ...

**6**

votes

**5**answers

294 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

**0**answers

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 ...