# Tagged Questions

**2**

votes

**1**answer

73 views

### Is first term of my cost function convex?

I have an optimization problem in the form of
[\begin{array}{l}
\mathop {{\rm{Minimize}}}\limits_{\bf{X}} \,\,\,2\left| \delta \right|\sqrt {{\rm{Tr}}\left( {{\bf{A}}{{\bf{X}}^2}} \right)} {\rm{ - ...

**0**

votes

**1**answer

63 views

### Relation between the subordinate norm and the spectral radius of a matrix

Let's define the following subordinate norm of a $(NM \times NM)$ matrix A norm as follows
\begin{eqnarray*}
||A||_{2,b} = \mathrm{max}_{x \in \mathbb{C}^{NM}} \left \{ \frac{||A x||_b}{||x||_2} ...

**3**

votes

**1**answer

136 views

### Alike looking matrices imply convergence of eigenvalues?

This is a question about convergence of eigenvalues which essentially came up in studying the spectrum of St.-Liouville operators.
We want to look at matrices that agree in most of their entries and ...

**3**

votes

**1**answer

85 views

### On the solution of a generalized Lyapunov equation

We shall reconsider the following equation
$$X=F_{1}XF_{1}^{T}+...+F_{p}XF_{p}^{T}+C$$
where $p$ is a positive integer and $C$ is a known symmetric positive semidefinite matrix.
I met with this ...

**3**

votes

**1**answer

102 views

### weak version of a Baez-Crans 2-vector space?

Baez and Crans defined a 2-vector space to be a category internal to the category of vector spaces (say over the reals). I am interested in categories that are equivalent to Baez-Crans vector spaces ...

**0**

votes

**0**answers

63 views

### Relation between the block maximum norm and the Euclidean norm of a matrix

I am trying to give answer to the following question:
Let's define de block maximum norm of a $N*M \times N*M$ matrix as
\begin{eqnarray*}
\parallel A \parallel_b = max_{x \neq 0} [ \parallel A x ...

**0**

votes

**0**answers

101 views

### Equivalence of Positive Matrix in Infinite Dimensional Vector Space

What is the corresponding linear operator on an infinite dimensional vector space, say a Banach space or Hilbert space, to the nonnegative matrix on a finite dimensional vector space? What is the ...

**4**

votes

**1**answer

444 views

### Die hard nilpotent spaces

Let $V\subset\mathbb{C}^{n\times n}$ be a linear space consisting of $n\times n$ complex matrices. Say that $V$ is nilpotent if every matrix $v\in V$ is nilpotent; denote by $V^k$ the subspace spanned ...

**2**

votes

**2**answers

146 views

### Matrix, singular values, Moore-Penrose-pseudoinverse

If A is any real mxn-matrix consider the block matrix
$\begin{pmatrix} E&A^T \\ A&0\end{pmatrix}$. This matrix seems to have close connections with pseudo inverse, svd etc. which are probably ...

**0**

votes

**0**answers

48 views

### Finding gradient of an optimization

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me?
Assume that we have an optimization ...

**0**

votes

**1**answer

63 views

### Integral over Kronecker product

Let $A : [0,T] \to \mathbb R^{n \times n}, t \mapsto A(t)$ be smooth with the property that
$$ \int_{0}^T A(t) dt $$ is invertible.
Does it then follow that the matrix
$$ \int_{0}^T A(t) \otimes ...

**4**

votes

**1**answer

76 views

### Sensitivity of the range of a matrix

The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal ...

**10**

votes

**1**answer

208 views

### Factor a sum of products of cofactors

Let $M$ be any $n\times n$ matrix.
We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row $i$ and column $j$ of $M$.
We can write ...

**7**

votes

**3**answers

384 views

### Diagonalization via the Toda flow

inAccording to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalized via the Toda flow. More specifically, if ...

**1**

vote

**0**answers

99 views

### Default Orientation of Vectors [closed]

When I started studying math in 1982 in Germany, there seemed to have been a change in the choice of the default orientation of vectors; while it was row-vectors till then, it changed to ...

**7**

votes

**1**answer

198 views

### Horn's inequalities for n matrices

Where I can find necessary and sufficient conditions on eigenvalues of Hermitian matrices with the relation $$A_1 + A_2 + ... + A_n = A_0 ,$$
i.e. Horn's inequalities for n matrices?
Can such ...

**2**

votes

**0**answers

48 views

### Reducing $\ell_1$ norm of non-full-rank matrices

I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ...

**9**

votes

**1**answer

283 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 ...

**5**

votes

**3**answers

162 views

### Norm of triangular truncation operator on rank deficient matrices

Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$
It is known that for arbitrary $A_{n\times n}$
$$\|T\circ ...

**2**

votes

**2**answers

108 views

### Boundedness of ratio of linear functions

Consider the function
\begin{eqnarray}
f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i},
\end{eqnarray}
over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...

**1**

vote

**1**answer

143 views

### Linear map with two “incompatible” representations

Let $K$ be a field and let $V$ be the set of sequences $\{v_1,v_2,\dots\}$ of elements of $K$. If $A=\{a_1,a_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ ...

**3**

votes

**1**answer

146 views

### What is a degenerate Legendre Transformation?

I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...

**0**

votes

**0**answers

48 views

### Uniqueness of a quadratic time-dependent matrix equation

Let $v: [0,1] \to \mathbb R^n, t \mapsto v(t)$ continuously differentiable with the property that for any constant vector $h \in \mathbb R^n$ the fact that $v(t)^{\top} h = 0$ for all $t \in [0,1]$ ...

**3**

votes

**1**answer

253 views

### Number of Matrices with bounded determinant

Here's my question:
Let $k,B,C$ be positive integers such that $B<C$. Can you give an upper bound for the number of $k\times k$ integer matrices having entries bounded in modulus by $B$ having ...

**4**

votes

**3**answers

219 views

### Is this function well studied?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Define the simplex
\begin{align}
\mathcal{S}=\left\{[x_1,\dots,x_L]\mid x_i\geq 0,~\sum_{i=1}^{L}x_i=1 \right\}
\end{align}
and consider the ...

**4**

votes

**1**answer

158 views

### Isomorphism of matrix ring over ore domain

Let $R_1,R_2$ be (left and right) ore domains. Does $ Mat_n(R_1)\cong Mat_m(R_2)$ implie m=n and $q.f.(R_1)\cong q.f.(R_2)$?
An counter example, a proof or a reference is welcomed.
Thanks

**6**

votes

**2**answers

271 views

### elementwise functions of positive definite matrix

The fact that the Schur (that is, element wise) product of two positive definite (symmetric) matrices is positive definite immediately implies (using the convexity of the positive semi definite cone) ...

**7**

votes

**1**answer

158 views

### Finite-dimensional inverse limits of double-dual spaces

Let $k$ be a field and $\{V_i\}_{i \in I}$ a filtered projective system of $k$-spaces with transition maps $f_{ji}: V_j \rightarrow V_i$ for $i \leq j$ (for my purposes we may assume the index set is ...

**1**

vote

**1**answer

192 views

### Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation:
$X=c \cdot AXA' - diag(c \cdot AXA')+ I$,
where
(1) $A \in R^{n \times n}$ is a given matrix whose element ...

**2**

votes

**1**answer

98 views

### Comparison of the smallest eigenvalues of two tridiagonal matrices

Let $n\geq2$ be an integer and $E_{ii}$ for an integer $2\leq i\leq n$ be the $n\times n$-matrix with its $ii$-entry equal to 1 and remaining entries equal zero. Furthermore, let ...

**10**

votes

**3**answers

586 views

### Are all vector-space valued functors on sets free?

Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between ...

**0**

votes

**1**answer

76 views

### Solution of infinite dimension linear system

Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$.
For fix n,
we can construct n dimension linear equation ...

**0**

votes

**0**answers

144 views

### Cholesky decomposition of a large covariance matrix

I have a tricky problem concerning a covariance matrix cholesky decomposition.
What I need is to obtain the cholesky decomposition of the estimated variance matrix of the set of samples stored in a ...

**0**

votes

**3**answers

212 views

### Matrix $A$ such that for all matrices $B$ the product $AB$ has a row with not a single zero

Let $A$ be a given fixed $n \times m$ matrix. We also consider matrices $B$ of dimension $m \times p$. I am interested in those matrices $A$, for which for all $B \in \mathbb R^{m \times p}$ with ...

**0**

votes

**0**answers

37 views

### A Optimization problem using co-ordinates of joint numerical range.

Let $\mathbf{A}_1,\dots,\mathbf{A}_L$ be $N\times N$ hermitian matrices. Define the mapping from the $N-$dimensional unit sphere to $\mathbb{R}^L$ as
\begin{align}
...

**4**

votes

**1**answer

653 views

### Surprising connection between linear algebra and graph theory

What is the intuition for linear algebra being such an effective tool to resolve questions regarding graphs?
For example, one can determine if a given graph is connected by computing its Laplacian ...

**9**

votes

**2**answers

197 views

### A differentiable one-parameter family of codimension 2 subspaces of $\mathbb{C}^n$ cannot fill $\mathbb{C}^n$, right?

Suppose that $P(t)$ is a one-parameter family of rank 2 self-adjoint projections on $\mathbb{C}^n$ that vary analytically in the real parameter $t \in [0,1]$. I claim that there must exist a vector $x ...

**2**

votes

**2**answers

227 views

### Finding the set of all $0-1$ vectors in an affine subspace

We are given a $0-1$ matrix $A$ with constant row and column sum, and we need to find out if there exists a $0-1$ vector in the solution space of $Ax = \mathbf{1}$ over $\mathbb{Q}$ (or $\mathbb{Z}$) ...

**3**

votes

**4**answers

269 views

### Determinant of sum of Kronecker products

Given four real symmetric matrices $A,B \in \mathbb{R}^{n \times n}$ and $C,D \in \mathbb{R}^{m \times m}$, is there an efficient way to compute the determinant:
$\det|A \otimes C + B \otimes D |$

**1**

vote

**0**answers

25 views

### the 3th and 4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf{z}} \in {\mathbb{C}}^{n \times 1}$ is a CSCG random vector denoted with $\mathcal{C} ~ (\bf{\mu} _0,\bf \Sigma _0)$ where $\mu _0$ and $\bf \Sigma _0$ are mean and contrivance ...

**1**

vote

**0**answers

83 views

### What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$
$$
F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2)
$$
on the vector space ...

**0**

votes

**2**answers

107 views

### Matrix equation solving guidelines [closed]

Does anyone how to solve this matrix equation?
$$ PXQ^T+P^TXQ=A$$
where all matrices are real and square. Can you provide me with some guidelines?
Thanx

**15**

votes

**1**answer

294 views

### A possible extension of a determinant inequality

It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$
I am considering a possible extension of this result. Let $\mathbb{M}_m(\mathbb{M}_n)$ ...

**1**

vote

**1**answer

296 views

### Under what conditions there is a one-to-one mapping between a product of matrices and the sequence of matrices leading to the product?

I have a set of matrices $A_1,\ldots,A_n$. Let $\mathcal{A} = \{A_i\}$.
What are some simple conditions under which for any sequence of indices between $1$ and $n$, $a_1,\ldots,a_m$, the product
...

**0**

votes

**1**answer

160 views

### Positive Semidefinite matrix [closed]

Let $A$ be an $n\times n$ symmetrix matrix, if $\forall i$,
$a_{ii}\geq |a_{ij}|,\forall j$
satisfies, can we say that $A$ is a positive semidefinite matrix? I tried to find a counter example, but ...

**7**

votes

**1**answer

209 views

### On an inequality among determinants

For Hermitian matrices $X, Y$, I write $X\ge Y\ge 0$ to mean $X-Y$ and $Y$ are positive semidefinite.
In Lemma 2.5 of [Linear Algebra Appl. 452 (2014) 1-6] I proved that if $X + Y\ge W + Z$,
$X\ge ...

**2**

votes

**0**answers

26 views

### L is the laplacian matrix of an undirected graph, D is a diagonal matrix. What does the cofactor of L+D looks like?

We know that any cofactor of the Laplacian matrix constant, and equal to the number of spanning tree. How are the cofactors change if I just add a diagonal matrix to the Laplacian matrix.
Any help ...

**2**

votes

**0**answers

111 views

**19**

votes

**0**answers

644 views

### conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...

**3**

votes

**1**answer

91 views

### Space of matrices B for which there is a solution to Bx=c for a given c

Let $F$ be a field, $k$ and $m$ natural numbers with $k \leq m$, and $c \in F^m$.
Is there some name for the set $\mathcal{B}_c = \{ B \in F^{m \times k}\, | \,\, \exists x \in F^k $s.t. $ Bx = c\}$ ...