A property of an object is called invariant if, given some steps that alter the object, always remains, no matter what steps are used in what order.

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For general non-symmetric square matrices is there a matrix norm that is invariant under similarity transformations?

I think that there is no similarity-invariant matrix norm for general matrices. But are there similarity invariant norms for special types of matrices (e.g. for matrices whose eigevalues are different ...
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Invariant measures for stochastic processes

I have some doubts about the concept of invariant measure for a stochastic process. Let me introduce a definition. Given $(\Omega, \mathcal{E}, \mathbb{P})$ a measure space, $H$ Hilbert space, let be ...
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Is it possible to find a 2D distribution function such that the higher order moments always exist?

Is it possible to generate a 2D distribution function $f(x,y)$ with function supports specified as $[-a,a]$ and $[-b,b]$ for $x$ and $y$ respectively, such that it always has moments which are NON ...
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A Unique Invariant subspace for a set of matrices

Im wondering if anyone can give me a good reference or answer this question which may have already be solved. For a set of generic $n\times n$ matrices $A_1,A_2,...,A_k$, such that they share only ...
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Proving a $k$-multilinear symmetric map is invariant iff a condition is satisfied

In Huybrecht's book on complex geometry, he states the following lemma on page 193: Lemma 4.4.2: The $k$-multilinear symmetric map $P$ is invariant if and only if for all $B,B_1,\ldots, B_k \in ...
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Write $V=P_2(\mathbb{R})$ as a direct sum of $V=W_1\oplus W_2 \oplus W_3$

So, if I let $T:P_2(\mathbb{R}) \rightarrow P_2(\mathbb{R})$ and is a linear endomorphism given by $T(f(x))=f(x)-f(2x-1)$. Then I have to write$V=P_2(\mathbb{R})$ as a direct sum of $V=W_1\oplus W_2 ...
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306 views

How to prove that the Kronecker delta is the unique isotropic tensor of order 2?

Is there a way to prove that the Kronecker delta $\delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = \lambda \delta_{ij}$ ...
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Prove a transformation is a variational symmetry for J

The following problem is from The Calculus of Variations by B.von Brunt (page 215, Exercise 9.2.1) Let $$ J(y)=\int_a^b xy'^2\mathrm{d}x. $$ Show that the transformation $$ X=x+\epsilon2x\ ...
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A question about Invariant subspaces of an algebra.

I feel that this is a very simple problem, but somehow I don't see the solution. I want to show that if $A$ is a subalgebra of $B(H)$ containing $1$ then if $B\in SOTcl(A)$, for every n, ...
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Invariants of point sets in an affine space

A distance between a pair of points in an affine space is invariant under translation, rotation and reflection. An angle in a triangle whose corners are tree points is also invariant under scaling. ...
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notation for invariation

Let $\Lambda = \{T \in \operatorname{Her}_2(\mathcal{O}) ; T \ge 0\})$ and $\mathcal{O}$ the maximal order of some quadratic imaginary number field. I write $T[U] := U^* \cdot T \cdot U$ where $U$ is ...
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What should you call a property, like an invariant, but that is reversed instead of preserved?

Suppose $P$ is some property of some objects and $f$ is a function on those objects. If $Px$ implies $Pf(x)$ and $\lnot Px$ implies $\lnot Pf(x)$, then we might say that "$P$ is invariant under $f$". ...
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Gaussian Distribution Under Orthogonal Transformation

Let $\mathbf{H}\in \mathbb{R}^{n\times n}$ be a random matrix whose every element has a Gaussian distribution with mean $m_{ij}$ and variance $\sigma^2$ i.e. ...
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Finding an invariant polynomial under a matrix action

I asked this question as a mathematica question: http://mathematica.stackexchange.com/questions/41689/finding-a-certain-invariant-polynomial-using-matrix-coordinates but maybe it will get more ...
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Time-invariant Differential Equation

I was wondering if it is safe to assume that $$ (1/t)y'(t) + (1/t)y(t) = 0 $$ is a time-varying differential equation. I would say these (1/t) factors cancel out and make the eq. time invariant. ...
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A set is partitioned into $k$ non-empty sets, and the difference between elements of each subset is given, reconstruct each subset

We have a set $A = \{1, 2, 3, \dots, n\}$. This set is partitioned into $k$ non-empty subsets: $$A_1 = \{a_1, a_2, \dots, a_{m_1}\}$$ $$A_2 = \{a_{m_1 + 1}, a_{{m_1} + 2}, \dots, a_{m_2}\}$$ $$.$$ ...
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finding invariant subgroups under all automorphisms

I'm trying to find group G s.t every subgroup of G is invariant under all automorphisms, or conditions for G. For example; cyclic groups and simple groups have this condition.
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How to show that the Harris corner detector is rotation invariant

I understand what it means for the Harris corner detector to be rotation invariant. But what steps do you need to take to show that it is rotation invariant?
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Calculate the variance from a function of normal random variable

I am new to the topic that I found difficulty for the question: Given the function $g(x) = e^{-X}$, $X \sim N(0,1)$, calculate the variance of $g(x)$. I know the answer is $e(e-1)$. But I don't ...
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$S_k$ action on $A/I$

Let $S_2$ be a finite group of order $2$ and let $S_2$ act on $k[x,y]$ by interchanging $x$ and $y$, where $k=\overline{k}$. Then since $$ R = \left( \dfrac{k[x,y]}{(x+y)} \right)^{S_2} = ...
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Invariant set: sufficient condition

Given a system $\dot x = f(x)$, $x \in \mathbb{R}^n$ with a smooth $f(x)$. Let $D$ be a set in $\mathbb{R}^n$ with a smooth boundary $\partial D$ such that $\left.\langle f(x), n(x) \rangle ...
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Matrix properties invariant under scalar multiplication

Given a square real matrix $A\in M_n(\Bbb R)$, what are ALL the properties invariant under scalar multiplication? In other words: which are the properties shared by all the $\lambda A$'s when ...
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Checking hand manipulations of matrices

Beginning with a 4*3 matrix: 5 4 -1 2 3 -3 3 4 -4 1 3 -2 I have to perform four manipulations on it, which I did by hand. I wanted to ask if my thinking and/or ...
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Iterative function eventually reaching identity

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that ...