All Questions
6,292 questions
1
vote
0
answers
100
views
Changing a nonlinear equality constraint into some conic inequality plus rank constraint
If we have a constraint optimization problem in which one of our constraint is $\prod\limits_{k = 1}^N {\left( {x - {a_k}} \right) = 0} $ . How could this nonlinear equality condition be changed into ...
- 33
2
votes
1
answer
135
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{ - ...
- 33
0
votes
1
answer
836
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} \...
2
votes
1
answer
346
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 ...
user54300
3
votes
1
answer
587
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 ...
- 77
4
votes
1
answer
190
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 ...
- 1,710
5
votes
1
answer
611
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 ...
user56203
2
votes
2
answers
557
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 ...
- 21
0
votes
0
answers
80
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 ...
- 161
0
votes
1
answer
456
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
$$ \int_{0}^T A(t) \otimes A(...
user45183
4
votes
1
answer
161
views
Sensitivity of the range of a matrix
The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal ...
- 20.2k
10
votes
1
answer
837
views
Factor a sum of products of cofactors
Let $M$ be any $n\times n$ matrix.
We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row $i$ and column $j$ of $M$.
We can write ...
- 2,902
11
votes
3
answers
1k
views
Diagonalization via the Toda flow
According to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalised via the Toda flow. More specifically, if $X=X(t)\in\...
- 2,933
1
vote
0
answers
187
views
Default Orientation of Vectors [closed]
When I started studying math in 1982 in Germany, there seemed to have been a change in the choice of the default orientation of vectors; while it was row-vectors till then, it changed to column-...
- 13.2k
9
votes
1
answer
401
views
Horn's inequalities for n matrices
Where I can find necessary and sufficient conditions on eigenvalues of Hermitian matrices with the relation $$A_1 + A_2 + ... + A_n = A_0 ,$$
i.e. Horn's inequalities for n matrices?
Can such ...
- 1,318
3
votes
0
answers
128
views
Reducing $\ell_1$ norm of non-full-rank matrices
I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ${\...
- 291
10
votes
1
answer
938
views
Why does this antisymmetric product factor out a determinant?
Consider a generic $n \times n$ matrix $M$.
Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$:
$$R_q = M_q^T (M_q ...
- 2,902
5
votes
3
answers
693
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 $A_{n\times n}$
$$\|T\circ A\|\le\frac{\...
- 153
2
votes
2
answers
219
views
Boundedness of ratio of linear functions
Consider the function
\begin{eqnarray}
f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i},
\end{eqnarray}
over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; \...
- 23
1
vote
1
answer
176
views
Linear map with two "incompatible" representations
Let $K$ be a field and let $V$ be the set of sequences $\{v_1,v_2,\dots\}$ of elements of $K$. If $A=\{a_1,a_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ $$...
- 6,924
4
votes
1
answer
788
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 ...
- 245
0
votes
1
answer
1k
views
How do eigenvalues change if we duplicate a row and column of a symmetric matrix?
Let $\bf A$ be a $n \times n$ symmetric positive semidefinite matrix whose first column is denoted by ${\bf a}_1$. We define a new matrix,
$$ {\bf B} = \begin{bmatrix} a_{11} & {\bf a}_1^T \\ {\bf ...
3
votes
1
answer
320
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 ...
- 31
4
votes
3
answers
308
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 ...
- 1,421
4
votes
1
answer
477
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
- 53
12
votes
3
answers
3k
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) ...
- 96.4k
7
votes
1
answer
462
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 $...
- 409
2
votes
2
answers
607
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 ...
- 241
2
votes
1
answer
430
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 $H_n:=\mathrm{tridiag}...
- 21
14
votes
3
answers
1k
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 ...
- 3,937
0
votes
1
answer
546
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 ...
- 111
2
votes
0
answers
1k
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
315
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 non-...
user45183
0
votes
0
answers
81
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}
\mathcal{S}=\{\left(\mathbf{u}^H\...
- 1,421
13
votes
1
answer
5k
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 ...
- 407
9
votes
2
answers
274
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 ...
- 630
4
votes
2
answers
383
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}...
- 1,197
4
votes
4
answers
3k
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 |$
- 143
1
vote
0
answers
46
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 matrix,...
- 275
2
votes
0
answers
123
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)
$$
on the vector space $V^{\mathbb{R}}:={\...
- 14.2k
0
votes
2
answers
163
views
Matrix equation solving guidelines [closed]
Does anyone how to solve this matrix equation?
$$ PXQ^T+P^TXQ=A$$
where all matrices are real and square. Can you provide me with some guidelines?
Thanx
11
votes
2
answers
802
views
Functions that Calculate their $L_p$ Norm
are there any examples of functions $f:x\in\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ and intervals $(a,b), 0\le a \lt b \le \infty$ , for which $$\Big(\int_a^b{|f(x)|^p dx}\Big)^\frac{1}{p} = f(p)$$ $$\...
- 13.2k
19
votes
1
answer
856
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 $\mathbb{M}_m(\mathbb{M}_n)$ ...
- 1,748
1
vote
1
answer
386
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? [closed]
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
$$\...
- 131
1
vote
1
answer
343
views
Positive Semidefinite matrix [closed]
Let $A$ be an $n\times n$ symmetrix matrix, if $\forall i$,
$a_{ii}\geq |a_{ij}|,\forall j$
satisfies, can we say that $A$ is a positive semidefinite matrix? I tried to find a counter example, but ...
- 187
8
votes
1
answer
256
views
On an inequality among determinants
For Hermitian matrices $X, Y$, I write $X\ge Y\ge 0$ to mean $X-Y$ and $Y$ are positive semidefinite.
In Lemma 2.5 of [Linear Algebra Appl. 452 (2014) 1-6] I proved that if $X + Y\ge W + Z$,
$X\ge W\...
- 1,748
3
votes
1
answer
272
views
If L is the laplacian matrix of an undirected graph, and D is a diagonal matrix, what does the cofactor of L+D look like?
We know (e.g. [Godsil, Royle: Algebraic Graph Theory, Lemma 13.2.3]) that any cofactor of the Laplacian matrix of a graph is constant, and is equal to the number of spanning trees of the graph. How do ...
- 31
2
votes
0
answers
349
views
An integral problem related to matrix determinant
I am stuck in an integral problem:
$$\int_{\mathbb{R}^d}(\det(\mathbf{A}+k(\mathbf{x}-\mathbf{y}_0)(\mathbf{x}-\mathbf{y}_0)^T))^{-r_1}\prod_{i=1}^n(\det(\mathbf{B}+(\mathbf{x}-\mathbf{y}_i)(\mathbf{...
- 65
24
votes
0
answers
1k
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 ...
- 483
3
votes
1
answer
109
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\}$ ...
- 33