**0**

votes

**0**answers

2 views

### Elementary bound on operator norm on symmetric tensors: reference request

My education didn't really cover Tensors very well, so I'm getting stumped by a quite elementary question.
Let $T_k$ be a type k symmetric tensor. We can define the "operator norm" (or the induced ...

**-3**

votes

**0**answers

17 views

### linear algebra, existense of polynomials [on hold]

True or False? How to prove it?
There do not exist polynomials p(t) and q(t), and scalars a, b, c, d, e, f , for which the following equations hold for all t:
(Hint: use Theorem 9.)
ap(t) + bq(t) = 1 ...

**13**

votes

**1**answer

205 views

### Minimum value of $|p(1)|^2+|p(2)|^2 +…+ |p(n+3)|^2$ over all monic polynomials $p$

Let $n$ be a positive integer. Determine the smallest possible value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$ of degree $n$.
This question was proposed (problem ...

**0**

votes

**0**answers

23 views

### Column Inner Products vs. Row Inner Products

Given two matrices $A,B\in\mathbb{R}^{n\times r}$ where $A$ has orthogonal columns and $A^TB$ is symmetric, are there any non-trivial interesting relationships / inequalities between the following ...

**0**

votes

**1**answer

166 views

### Worst case difference in rank by column-row swapping

Given a matrix $m\in\{-1,+1\}^{n\times n}$. Consider $m^\sigma$ to be collection of all matrices obtained from $m$ by permuting rows and columns.
Consider $\mathscr{M}[m^\sigma]$ to be collection of ...

**28**

votes

**3**answers

2k views

### A curious determinantal inequality

In my study, I come across the following curious inequality, which I do not know a proof yet (so I am asking it here).
Let $A, B$ be $n\times n$ (Hermitian) positive definite matrices. It is very ...

**12**

votes

**4**answers

659 views

### Determining if a matrix is orthogonal

Let g be an element of $GL_n(\mathbb C)$. We know that there are orthogonal groups $O(\beta)=\{X\in GL_n(\mathbb C) \mid X^t\beta X=\beta\}$ for any $\beta$, invertible symmetric matrix. Though these ...

**0**

votes

**0**answers

25 views

### Isotropic subspaces in a symplectic vectorspace over $GF(q)$

Let $V$ be a symplectic vectorspace of dimension $2n$, and $r\mid n$. Is this statement true?"There is an isotropic spread of $r$ dimensional subspaces in $V$". By an isotropis subspace I mean a ...

**2**

votes

**1**answer

185 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 ...

**0**

votes

**1**answer

84 views

### A question about the Vandermonde determinant

We know that the Vandermonde determinant of order $n$ is the determinant defined as follows:
$$\begin{vmatrix}
1&x_1&x_1^2&\dots&x_1^{n-1}\\
...

**2**

votes

**1**answer

93 views

### Computation Time of Smith Normal Form in Maple

I am using maple to compute the Smith Normal Form of a matrix of size 120*120 and it seems that I will never get an answer back. I have checked my code for small cases and I believe that it is ...

**11**

votes

**1**answer

324 views

### Probability that random nonnegative integer matrix is singular

Q. What is the probability that an $n \times n$ matrix, whose elements
are independent uniformly random integers in $\{0,1,\ldots,k\}$, is singular?
For example, for $n=3$ and $k=2$, the first ...

**3**

votes

**2**answers

124 views

### Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case
$A = \begin{pmatrix} ...

**0**

votes

**0**answers

54 views

### Matrix representation

Let $\mathbf{c}\in \mathbb{R}^n$ and
$\mathbf{X}(s)= \begin{bmatrix} X_{11}(s) & X_{12}(s) & \cdots & X_{1n}(s) \\ X_{21}(s) & X_{22}(s) & \cdots & X_{2n}(s) \\ \vdots & ...

**2**

votes

**1**answer

80 views

### Shared maximum eigenvector

Let us consider two arbitrary Hermitian square matrices $\mathbf{A,B}$ with the same dimension. Given $\mathbf{v}$ the eigenvector associated to the maximum eigenvalue of $\mathbf{A}$:
Are there ...

**5**

votes

**1**answer

218 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

**1**answer

48 views

### Sum of two parts of a continuous stochastic process

Let $X$ be a centered continuous stochastic process which is square integrable on $[0,2]\times \Omega$ and the basis of $L^2(0,2)$ is $\{e_i\}$. By using Karhunen-Leove Theorem one can write for all ...

**0**

votes

**0**answers

23 views

### Writing eigen functions of one Stochastic Process in terms of the eigen functions of another

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...

**3**

votes

**1**answer

171 views

### Largest symmetric matrix given rank

Let $\mathscr{M}[n,d]$ be collection of $n\times n$ symmetric matrices with real entries from $\{0,1\}$ such that every row/column is distinct with sum of every row/column being $d$.
What is minimum ...

**0**

votes

**0**answers

36 views

### Largest Set of Special Unitary Matricies With Invariant Subspace For Adjoint Action

I am trying to solve the following. Given the special unitary group $SU(n)$ and its adjoint action $Ad_{U}: \mathfrak{su}(n) \rightarrow \mathfrak{su}(n)$, what is the largest subset of $SU(n)$ such ...

**2**

votes

**1**answer

42 views

### Vanishing Restricted Isometric Constant

In compressed sensing, we are interested in the restricted isometry property. Suppose the design matrix is $n$ by $p$, consisting of $np$ iid $\mathcal{N}(0, 1/n)$ entries. Assume both $n$ and $p$ are ...

**0**

votes

**0**answers

63 views

### Issue of Tikhonov regularization and Sobolev spaces [closed]

This question related to the link below:
http://math.stackexchange.com/questions/1272235/space-of-tikhonov-regularization-of-an-ill-poised-problems
Tikhonov regularization gives smoothing results ...

**3**

votes

**0**answers

90 views

### Can I find the gap between the two least eigenvalues of this special matrix A(t)?

I am interested in finding the gap between the two least eigenvalues of $A(t)$, a Hermitian $N\times N$ sparse matrix whose diagonal elements are $a_it+b_i\,(1\leq i\leq N)$, and all off-diagonal ...

**0**

votes

**1**answer

300 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

**5**answers

4k views

### Eigenvalues of Symmetric Tridiagonal Matrices

Suppose I have the symmetric tridiagonal matrix:
$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\\
b_{1} & a & b_{2} & & ... \\\
0 & b_{2} & a & ... & 0 ...

**3**

votes

**0**answers

51 views

### Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same

I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, ...

**2**

votes

**1**answer

30 views

### Information on special matrices similar to Jacobi matrices

Jacobi matrices are well known and deeply investigated mathematical objects from various point of view. One can arrive at these operators while studying discrete systems of particles interacting with ...

**2**

votes

**3**answers

314 views

### Info on Symplectic/Orthogonal groups of Gl(n,R); R a ring, not necessarily division ring.

Hi, Everyone:
Does anyone know anything about orthogonal and symplectic groups
associated to Gl(n,R)? I am using symplectic/orthogonal in what I think
is the standard sense; I mean, we have an ...

**2**

votes

**2**answers

155 views

### Permutation covering of a $G$-lattice

Let $G$ be a finite group.
By a $G$-lattice we mean a finitely generated free abelian group $L$ with an action of $G$.
We say that $L$ is a permutation $G$-lattice if $L$ has a ${{\mathbf{Z}}}$-basis ...

**0**

votes

**3**answers

125 views

### Simple Spectrum of Jacobi matrices

I want to call a matrix a Jacobi matrix (cause there may be different notions of Jacobi matrices) if it is a tridiagonal matrix with positive off-diagonal entries. Now, I read that the spectrum of ...

**0**

votes

**0**answers

47 views

### Characterisation of vector fields solution to a simple equation

This question is complementary to another question I asked on math.stackexchange. I believe it is more subtle than it seems - it will become clearer when I provide more context - and probably hides ...

**2**

votes

**1**answer

186 views

### Classes of finite resoluble groups which are (faithfully) representable by triangular matrices?

Let $G$ be a group, $k$ a field and $T(n,k)\subset Gl(n,k)$,
the group of invertible upper triangular $n\times n$ matrices.
I know that if $\rho : G\rightarrow T(n,k)$ is faithful
(i.e. into) then ...

**0**

votes

**1**answer

233 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
& & & & ...

**8**

votes

**5**answers

560 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.

**1**

vote

**1**answer

119 views

### Prove or disprove a matrix logarithm equation

Let $\Bbb{S}_{++}^n$ denote the space of symmetric positive definite (SPD) $n\times n$ real matrices, and let $A,B\in\Bbb{S}_{++}^n$.
Is it possible to express the logarithm of $A^{-1}B$ as a ...

**9**

votes

**3**answers

284 views

### Real and Quaternionic Representations according to Weights

According to this question, it is easy to know whether a representation is self dual or not: just check if the weight distribution in space is symmetric about the origin.
Now, for self dual ...

**0**

votes

**0**answers

57 views

### Minimum rank of certain matrices

Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct.
What is minimum real rank of matrices in ...

**1**

vote

**0**answers

59 views

### Volume element of symmetric definite matrices in polar coordinates

I have a difficulty to understand the following statement. I don't ask for a proof but just understand the statement concretely (what it does mean, how to apply it...)
Let $\mathcal P_n$ be the ...

**44**

votes

**7**answers

3k 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

31 views

### Products of random permutations with fixed matrix

This question originates from an engineering problem, which I am solving. Any related references are highly appreciated.
Let $M_k(T)=\prod_{t=1}^T P_t S_k$ over some field (finite or reals), where ...

**1**

vote

**0**answers

195 views

### On matrix rank inequality

Let $A$ be a $\{0,1\}$ square matrix.
Let $J$ be all $1$ matrix.
Let $\bar{A}=J-A$.
Is it possible for $rk_+(A)\geq c\cdot rk_+(\bar{A})^d-1$ and $rk_+(\bar{A})\geq c\cdot rk_+({A})^d-1$ for some ...

**1**

vote

**0**answers

26 views

### Smooth Parameterization of Set of Linear Subspaces [migrated]

Consider a finite dimensional vector space $V$(with inner product if needed), for example $\mathbb{R}^n$, for $m<n$, I want to give a smooth parameterization of the set of $m$-dimensional subspaces ...

**9**

votes

**2**answers

286 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

35 views

### Spectral radius of a rank-1 perturbation

Suppose that $A$ is an $n \times n$ matrix, and $u$ and $v$ are vectors. The matrix determinant lemma lets us easily compute the determinant of $A+uv^T$, while the Sherman-Morrison formula gives us ...

**0**

votes

**1**answer

175 views

### Axiomatic explanation of why the volume of a parallelepiped is equal to the area of its base times its height [closed]

I asked this in MSE, it flashed and disappeared.
Let $V_n$ be the volume on the set of polytopes in $\mathbb R^n$, defined axiomatically, i.e. a functional, that assigns to each polytope ...

**1**

vote

**1**answer

61 views

### Determinant of a Certain Positive-Definite Block Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$?
$$\Gamma=\left( {\begin{array}{cc}
I & B \\
B^{*} & I \\
\end{array} ...

**3**

votes

**1**answer

130 views

### minimal polynomial of unipotents in orthogonal group

Consider split orthogonal group O(2l) over a field of characteristic zero. We may assume the matrix of bilinear form to be $\begin{pmatrix} O&I\\ I&0\end{pmatrix}$. Let u be a unipotent in ...

**2**

votes

**0**answers

104 views

### Determinant Evaluation

Is there a closed form (something involving a ratio of products) for:
$$\det\left[\binom{a_i+c}{a_i-i+j}\right]_{1\leq i,j\leq t},$$
where $a_i,c$ are positive integers? I think with $c=0$ this is ...

**6**

votes

**1**answer

114 views

### Monte-Carlo computation of the Smith normal form

Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed:
Suppose $P, ...

**27**

votes

**4**answers

873 views

### Why there is a relation among the second-order minors of a symmetric $4\times 4$ matrix?

A $4\times 4$ symmetric matrix
$$
\left(
\begin{array}{cccc}
a_{11} & a_{12} & a_{13} & a_{14} \\
a_{12} & a_{22} & a_{23} & a_{24} \\
a_{13} & a_{23} & a_{33} & ...