Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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0
votes
0answers
9 views

Is there some quantitative version of interlacing of eigenvalues of a matrix under rank-1 update?

Given a real symmetric matrix $A$ and a vector $v$ of the same dimension we know that the eigenvalues of $A + vv^T$ are left interlaced by the eigenvalues of $A$. But do we have any quantitative ...
-1
votes
0answers
34 views

how to solve a linear equation (Ax-b)T- lamda(c)? [on hold]

I'm trying to solve an linear optimization problem, it's first order Lagrange condition leas to this equation $$ (Ax-b)^T A- \lambda C = 0. $$ Here $A$ is an $m\times n$ matrix, $m>n$, $C$ ...
-4
votes
0answers
20 views

Show that the mapping A linear. Lays down rules for adjoint transformation A * [on hold]

Let V n-expansive real vector space with scalar product, a and b given linearly independent vectors from the space V. mapping A: V -> V is given by Regulations Ax = (x, a), * b Assign eigen values ...
2
votes
0answers
54 views

Condition number after preconditioning

Suppose $A$ and $P$ are symmetric, positive definite matrices and that we factor $P^{-1}=EE^\top.$ Is it true that the condition number of $PA$ is upper-bounded by the condition number of ...
-4
votes
0answers
18 views

How many are there orthogonal transformations? [on hold]

which transform the line x = y / 2 = z line -x / 2 = y = z and the line x = -y = with the line x = y = z? Find the matrix of any of them in a standard basis I have tried to equal equations.
-4
votes
0answers
27 views

system of linear equations [on hold]

These are the two known equations (I2+I3)-(I1+I4)/(I1+I2+I3+I4) = 2x/L (I2+I4)-(I1+I3)/(I1+I2+I3+I4) = 2y/L where I know x,y,L values. How can I find the values of I1,I2,I3,I4?
0
votes
1answer
49 views

Hadamard Product and Eigendecomposition

I just found this related question in here Q1. Given a positive definite matrix $\mathbf{A}$, consider its eigendecomposition $(\mathbf{A}\mathbf{V} = \mathbf{V}\mathbf{D})$. Consider an arbitrary ...
0
votes
0answers
19 views

Tensor Algebra Quotients of the Ext-Alg of an SU(2)-Module

Let $V$ be a simple (left) $SU(2)$-module, and $T(V)$ the tensor algebra of $V$. If we quotient $T(V)$ by the ideal generated by a simple submodule of $V \otimes V$, is there a general system for ...
1
vote
0answers
40 views

Epsilon-net of operator norm ball around Identity

Suppose I look at the set of matrices which are invertible and satisfy $$ \left\|A-Id\right\|_{op}<r $$ for some $r<1$, where $Id$ is the $n\times n$ identity matrix. An $\epsilon$-net of such ...
-1
votes
0answers
44 views

Characterize polytopes resulting from cutting a convex polytope by a hyperplane [on hold]

We have a convex polytope $P$ for which we know its set of vertices. Using this set we characterize the H-representation of $P$ as: $\mathbf{A}\mathbf{x} < \mathbf{b}$. If a hyperplane defined by ...
-4
votes
0answers
13 views

How to find the matrix of adjoint transformation R* according to the usual scalar product [on hold]

transformation R3 -> R3 is a" circle" around the line x / z = -y = z / 2
-3
votes
0answers
16 views

Calculate coordinate transformation matrix [on hold]

Calculate coordinate transformation matrix A in the base [1; x; x2]. In the space R2 [x] of real polynomials highest rate? D is given a scalar product. Sebi-adjoint transformation of A: R2 [x]! R2 ...
3
votes
1answer
54 views

Spectral theorem from Jordan decomposition in infinite dimensions

The finite-dimensional spectral theorem is a simple special case of the finite-dimensional Jordan decomposition. We also have various infinite-dimensional generalizations of the spectral theorem; can ...
3
votes
1answer
88 views

Eigenvectors as continuous functions of matrix - diagonal perturbations

The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
0
votes
0answers
23 views

On the induced norms of stochastic operator and its adjoint operator

The background: when studying the paper published in Automatica named '$H_{\infty}$ control and filtering of discrete-time stochastic systems with multiplicative noise' (volume 37, pp. 409-417), I ...
6
votes
2answers
190 views

What methods do we have to understand the spectrum of matrices with restricted entries?

Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"), What is the largest possible spectral radius of a $n \times n$ matrix ...
0
votes
0answers
44 views

Completing Karlin's proof of variation diminishing transformation theorem

In S Karlin's book total positivity there's a theorem that says if $K(x,y)$ is $TP_r$ (totally positive with degree $r$) and the sign change count of function $h$, $S(h) = n\leq r-1$, then ...
0
votes
1answer
68 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C. I'd like a function μ:Cn×n→[0,∞) ...
1
vote
1answer
33 views

Splines linearly independent

Let $N_1:=\chi_{[0,1]}$ be defined as this characteristic function and $N_n:=N_{n-1}*N_1$ then this leads to polynomials with support $[0,n]$. These splines are well-studied click for wikipedia My ...
6
votes
2answers
323 views

Is there a nice “synthetic” way for doing differential geometry on infinite dimensional vector spaces?

If $V$ is an infinite dimensional vector space, for example the space of smooth functions on $\mathbb{R}$, we can introduce some differential geometry concepts by choosing a topology on $V$ and doing ...
0
votes
0answers
11 views

How to treat non-identifiable states in Kalman filtering/dynamic linear models?

Let $x_t = G_tx_{t-1}+\omega_t$ with $\omega_t \sim \mathrm{N}(\mathbf{0}, \mathbf{W}_t)$ be a state equation and $y_t = F_tx_t+\nu_t$ with $\nu_t \sim \mathrm{N}(\mathbf{0}, \mathbf{V}_t)$ be a ...
4
votes
2answers
247 views

“Typical” convergence rate for the von Neumann mean ergodic theorem

The von-Neumann theorem states that for a unitary operator $U: {\cal H} \mapsto {\cal H}$, where ${\cal H}$ is a Hilbert space, the following holds: $$ \lim_{N\to \infty} \frac{1}{N} \sum_{n=1}^N U^n ...
5
votes
2answers
225 views

Minimum of squared sum minus sum of squares

I know that $$ \min_{\|x\|_2=1=\|y\|_2} \left(\sum_{k=1}^nx_ky_k\right)^2-\sum_{k=1}^nx_k^2y_k^2 \geq -1/2 $$ with equality whenever $|x_k|=\frac{1}{\sqrt{2}}=|y_k|$ for two coordinates. I'm ...
1
vote
0answers
45 views

non-intersecting families of subspaces

Given $V$, a vector space over a finite field $F$ of size $k$, if $\dim(V)=m$, and $r$ divides $m$, there exists a family of $r$-dimensional subspaces, whose size is equal to $(k^m-1)/(k^r-1)$ and ...
6
votes
1answer
190 views

Convexity of the product of two exponential matrices

Let $S\subset\mathbb{R}$ be a convex set and $\mathbb{S}^{n}$ be the set of real symmetric matrices of order $n\times n$. A matrix valued function $\Gamma: S \rightarrow \mathbb{S}^{n}$ is said to ...
1
vote
2answers
83 views

Nonlinear matrix differential equation

I want to solve the equilibrium of the following differential equation: $\dot{x_i} = \sum_j A_{ij} x_j + x_i \sum_j B_{ij}x_j$ which is essentiall in matrix notation: $\dot{\mathbf{x}} = ...
0
votes
0answers
63 views

Existing complete function space under suitable norm

This question was asked in math.stackexchange.com but no suitable answer was received, so I am posting it here. This is a question which came to me due to several previous question: sorry for the all ...
1
vote
0answers
77 views

What is the trace of this operator in $L^\infty$ (if this question make sense)?

Let $\Omega$ be a closed domain in $\mathbb{R}^N$, and $\lambda$ the corresponding Lebesgue measure. We define in $(L^\infty(\Omega), \Vert \cdot \Vert_\infty)$ the following operator : ...
1
vote
0answers
52 views

Smallest sum of original column entries in 2d matrix [closed]

I have an interesting optimization problem I am trying to solve now and I thought I'd share it here in order to find the best answer. The problem itself is not complicated and it is stated like this: ...
1
vote
1answer
38 views

Dimension of a certain subspace of univariate polynomials

Let $\mathbb{F}$ be an arbitrary field. For a polynomial $f\in\mathbb{F}[x]$, we use $Z(f)$ to denote set of roots of $f$ in $\mathbb{F}$. Let $S$ and $T$ be sets of elements of $\mathbb{F}$ of size ...
1
vote
1answer
55 views

Given $M$, minimize $|Mx|_0$

Given a matrix $M \in \mathbb{R}^{n \times m}$, I would like to find $\min_{x \in \mathbb{R}^m} \|Mx\|_0$ such that $x \neq 0^m$, where the $\ell_0$ "norm" is measured by simply counting the number ...
5
votes
1answer
320 views

How many random matrices does it take to generate a matrix algebra?

Let $\mathbb{F}$ be a finite field. Let $A\le \mbox{Mat}_n(\mathbb{F})$ be a matrix algebra. Is there a good bound on the number $k$ of random elements $a_1,\dots,a_k\in A$ that one needs to take ...
1
vote
0answers
48 views

Question about Eigenvalues of group elements [migrated]

The quaternion algebra is given by $\mathbb{H}$ = $\{a+bi+cj+dk \mid a, b, c, d \in \mathbb{R}, i^2 = j^2 = k^2 = -1, ij = k = -ji\} := \{z_1+z_2j \mid z_1, z_2 \in \mathbb{C}\}.$ I consider the ...
9
votes
0answers
199 views

Bunnity of multilinear maps

Is there a way to compute the following nullity of multilinear maps? As it is different from any nullity I know of, I call it bunnity after myself:-)) If it already has a name, it be nice to know it. ...
1
vote
1answer
1k views

Does this linear algebra construction based on a graph have a name, and where has it been studied?

In the paper Kochen-Specker set with seven contexts by Lisonek, Badziag, Portillo and Cabello, the following construction is used : Question : Have such constructions been used elsewhere, and if so ...
0
votes
0answers
43 views

Issue of Tikhonov regularization and Sobolev spaces

This question related to the link below: Shttp://math.stackexchange.com/questions/1272235/space-of-tikhonov-regularization-of-an-ill-poised-problems Tikhonov regularization gives smoothing results ...
0
votes
1answer
232 views

Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]

Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...
0
votes
0answers
62 views

Degree of permutation of hypercube

Given $S_0\cup S_1=T_0\cup T_1=\{0,1\}^n$, $S_0\cap S_1=T_0\cap T_1=\emptyset$, with $|S_i|=|T_i|$ for both $i\in\{0,1\}$, what is degree of transformation that simultaneously maps $S_i$ to $T_i$ for ...
2
votes
1answer
135 views

Linear independence of +/- 1 strings/vectors II

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 ...
1
vote
0answers
154 views

Is there a way to simplify this apparently huge characteristic polynomial calculation?

Say I am given the $0/1$ adjacency matrix of an undirected graph. Also I am given a representation $\rho$ of some group $G$ and an orientation has been arbitrarily chosen along each edge. Let ...
1
vote
1answer
117 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 ...
6
votes
4answers
431 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
1answer
210 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 $$ ...
2
votes
0answers
295 views

How to prove the following determinant identity? [migrated]

This problem is relevant to the spin operator matrix elements in the quantum 1D XY model. For any even integer $N$, define two sets ...
2
votes
3answers
346 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
1answer
93 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 ...
41
votes
7answers
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
0answers
38 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
1answer
37 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
0answers
163 views

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 ...