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.

learn more… | top users | synonyms

1
vote
3answers
45 views

Does a repeated eigenvalue always mean that there is an eigenplane under the transformation matrix?

If you have a 3x3 matrix, if you find that it has repeated eigenvalues, does this mean that there is an invariant plane (or plane of invariant points if eigenvalue=1)? I always thought that there was ...
-1
votes
2answers
15 views

Matrices for action wrt basis

Consider the permutation representation where $G=S_3$ on $\mathbb{C^3}$ with the action: $\pi(g)e_i=e_{g(i)}$ $W=\{ \lambda_i e_i ; \sum \lambda_i=0 \}$ is an invariant subsoace of vector space $V$ ...
0
votes
2answers
59 views

Invariant subspaces

Let $V$ is a finite dimensional vector space over $\mathbb{C}$ and $T$ be a linear operator on $V$ . How to prove $T$ has an invariant subspace of dimension $k$ for each $k = 1,2, \ldots ,\text{dim}V$ ...
0
votes
1answer
41 views

Invariant subspaces and dimentions

Let V is a finite dimensional vector space over C and T be a linear operator on V . How to prove T has an invariant subspace of dimension k for each k = 1,2, . . . ,dimV .
2
votes
0answers
41 views

Isospectral transformation of ODE

Is there a transformation of coefficients of differential operators w/periodic coefficients on the real line $a_n(x+2\pi)=a_n(x)$, that preserves the eigenvalues of their monodromy matrices? ...
2
votes
2answers
57 views

Invariant subspaces

Let T be a linear operator on a finite dimensional vector space V over a field F such that every subspace of V is invariant under T then how to prove T is digonalizable ? Is the converse true ?
-1
votes
1answer
40 views

Compactness and positive invariance of set under flow of ODEs

Given a system of ODEs, $$x'=y$$ $$y'=x-x^3-y$$ $$x(0)=x_0$$ $$y(0)=y_0,$$ also given a set $S=\{(x,y):V(x,y)\le k, x>0\}$, $V(x,y)=-\frac{x^2}{2}+\frac{x^4}{4}+\frac{y^2}{2}$, where ...
0
votes
3answers
41 views

Positive invariance of a set under a system of ODEs

Given the system of ODEs, $$x'=x(1-x-y)$$ $$y'=y(x-1),$$ $Q=\{(x,y):x\ge 0, y\ge 0\}$, and $S=(x,y)\in Q:x+y\le k$, $k>1$, I need to show that $S$ is invariant under this system of ODEs. Attempted ...
0
votes
1answer
37 views

Invariance of sets for systems of ODEs

Given the system of ODEs $$x' = x(1-y)$$ $$y'=y(x-1),$$ let the set $Q=\{(x,y):x\ge 0, y\ge 0\}$. Explain why $Q$ is invariant for this system of ODEs. My explanation: If $x > 1$ and $y<1$ then ...
4
votes
6answers
215 views

Proof there is no way to chose signs to make sequential sum $1+2+3+\cdots+10$ even [closed]

I've figured that for the sum $$1+2+3+4+5+6+7+8+9+10=55$$ There is no way to chose the signs of the numbers to get an even sum. I'm really struggling to prove this and would appreciate some ...
0
votes
2answers
16 views

Which complex polynomials in 3 variables are $GL_3(\Bbb{C})$ invariant?

A polynomial $p(x,y,z) \in \Bbb{C}[x,y,z]$ is $GL_3(\Bbb{C})$-invariant if $$ \forall \sigma \in GL_3(\Bbb{C}): p(\sigma(x,y,z)) = p(x,y,z).$$ How to characterize the set of $GL_3(\Bbb{C})$ invariant ...
0
votes
1answer
37 views

Jumping on the Coordinate lattice grid

Mr. Fat moves around on the lattice points according to the following rules: From point (x, y) he may move to any of the points $(y, x), (3x, −2y), (−2x, 3y), (x+1, y+4)$ and $(x − 1, y − 4).$ Show ...
0
votes
0answers
7 views

Closed Under Linear Transformation vs Invariance - Formally Naming a property of a set

I have a set of points in $\mathbb{R}^2$ given as $G(t;z_0,\theta_0)$ where $z_0 \in \mathbb{R}^2$ and $\theta_0$ represents a specific orientation. I have proved that, $G(t;z_0,\theta_0) = z_0 + ...
0
votes
1answer
57 views

Show permutation representation is reducible, by finding G-invariant subspace [duplicate]

$(\pi,V) $ is the permutation representation of the symmetric group $S_5 $, $ V=C^5$ and the action of standard basis vectors of $ V$ is given by $\pi(\sigma)e_i=e_{\sigma(i)} $ for $\sigma\in S_5 $ ...
1
vote
2answers
61 views

$T$ Invariant subspace

Please help me to write a step by step solution to this problem: Let $T$ be a linear operator defined on a finite dimensional vector space $V$. If $W$ is a $T$ invariant subspace with $V = ...
1
vote
1answer
45 views

Solving a Chessboard problem using the Invariance principle

Problem Statement There is an integer in each square of an 8 x 8 chessboard. In one move, you may choose any 4 x 4 or ...
2
votes
0answers
31 views

Invariant equation

If I have the equation: $x=-$tan$x$ I can send $x \rightarrow -x$ and the equation doesn't change. If I define $x>0$ one of the possible ranges $x$ can take without solving the equation is ...
0
votes
1answer
35 views

Applying invariance principle on a problem on sequence of positive integers

The problem statement: Start with the positive integers 1,...,4n-1. In one move you may replace any two integers by their difference. Prove that an even integer ...
4
votes
2answers
59 views

Understading the integral form of a conservation law

When I think of a conservation law I think of a continuity equation like the following $$\partial_t \rho = -\nabla \cdot \vec j$$ But now I'm reading a book on electrodynamics (that's honestly a bit ...
1
vote
1answer
42 views

What is an invariant Simplex

Let $$\Delta=\{x_i\in\mathbb{R}^k_+: \sum_{i=1}^kx_i=1\}$$ where k is a positive integer. I've read in a book the following: The simplex $\Delta \subset \mathbb{R}^k$ can be shown to be invariant ...
0
votes
0answers
21 views

Invariance of stationary wavelet transform

Suppose we are given 64 points $x_1,\ldots ,x_{64}$ and divide them into two groups $x_1,\ldots, x_{32}$ and $x_{33},\ldots , x_{64}$. Then we apply stationary wavelet transform to both these groups ...
0
votes
0answers
111 views

Left invariant Vector Field on $S^2$

How intuitively look like all left invariant vector fields on this manifold: the 2 dimensional unit sphere $S^2$ with the smooth structure inherited from $\mathbb R^3$? Why all left invariant vector ...
3
votes
2answers
44 views

Are Bezier curves invariant under conformal mapping?

I've spent quite a bit of time on google trying to find information on whether or not Bezier curves are invariant under conformal mapping (i.e. a conformal mapping of all points on the curve is the ...
0
votes
1answer
46 views

$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}_i) = 0$

$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}) = 0$ in the image below (third and fourth line of the proof!). Why?
2
votes
2answers
66 views

loop invariant for simple algorithms

The following is an algorithm which finds the maximum value in a list of integers, and I want to prove that it is correct by using a loop invariant. ...
0
votes
0answers
31 views

arg min-invariance for norm of vectorfield under linear transformation

Given a vectorfield $\vec{F}(\vec{c}) \in \mathbb{R}^n$ which is a function of some parameters $\vec{c}$, what constraints must you have on a matrix such that when you act on the vectorfield the ...
2
votes
1answer
46 views

find an invariant

I've been reading about the use of invariants in contest math. I saw the following problem (in my own words): There are $N = 2n$ numbers placed on a circle. Then we increase two any consecutive ...
0
votes
1answer
22 views

Invariance Dealing with Infected Squares

Twelve 1x1 cells of a 10x10 square are infected. Two cells are called neighbors if they share at least one vertex (thus an inner cell has 8 neighbors). In one unit time, the cells with at least four ...
1
vote
1answer
49 views

Numbers written on a board

The numbers $1,2,...,n$ are written on a board ($n\in\mathbb N$). In each step we take any two numbers $a,b$, remove them, and write either $a-b$ or $a+b$ on the board. After $n-1$ steps there will be ...
0
votes
1answer
22 views

Invariance problem dealing with the sums of units digits

We may write all the digits from 1 to 9 in a row in any order we like, and then we write plus signs between some digits (as many plus signs as we like). Finally, we evaluate the obtained expression. ...
1
vote
1answer
45 views

Lyapunov invariant set for affine systems

Given a linear system $\dot{x}=Ax$ such that the real part of every eigenvalue of $A$ is less than $0$, Lyapunov's equation $A^T P + P A = -Q$ with $Q$ being any suitably sized positive definite ...
0
votes
1answer
45 views

Are similar complex matrices again similar when each is expressed as a real matrix?

We know that, relative to this ordered basis {$(1,0),(i,0),(0,1), (0,i)$}, we can express a 2x2 complex matrix mapping $C^2 -> C^2$ as a $4x4$ real matrix (representing the same transformation of ...
0
votes
0answers
88 views

Proving span of a complex eigenvectors is an invariant subspace

Following these notes:http://www.math.uwaterloo.ca/~jmckinno/Math225/Week11/Lecture3r.pdf We wish to prove validity of the following in bold: Theorem 9.4.2 Suppose that $λ = a + bi$, $b \neq 0$, is ...
0
votes
1answer
71 views

State machine scenario: finding invariant

Alice, Bob, and Charles want to evenly distribute a dozen doughnuts. Initially, Alice has 5, Bob has 3, and Charles has 4. However, they want to do it according to the following rules: 1) Bob may ...
0
votes
0answers
47 views

Invariant question: Fifteen Puzzle

The Fifteen Puzzle consists of sliding square tiles numbered $1...15$ held in a $4\times4$ frame with one empty square. Any tile adjacent to the empty square can slide into it. The standard initial ...
0
votes
1answer
20 views

Finding invariants

If in a given ecosystem there are 30 chameleons living on an island: 15 red, 7 blue, 8 green. When two of a different color meet, they both change into a third color (ie. if a red and blue meet, they ...
0
votes
0answers
15 views

Verification about group actions and “uniformity” of an action

I spent some time revisiting group actions this week. I was hoping to get someone to verify a seemingly straightforward claim. I also had a thought on how "uniform" an action is on a space. ...
0
votes
1answer
48 views

Specific matrix has no 2-dimensional invariant subspaces

I have the endomorphism $$ M = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} $$ of a real vector space $V$. Note that this matrix is nilpotent (with $M^3 = ...
2
votes
0answers
32 views

Attractor $A$ with neighborhood $V$ such that $f^N(V)\subset V$ and $A=\bigcap_{n\in\mathbb{N}}f^n(V)$, then $\omega(x)\subset A$ for all $x\in V$?

Let $f\colon X\to X$ be a continuous map on a (compact) topological space $X$. Let $A$ be an attractor for $f$, i.e. there is a neighborhood $V$ of $A$ such that $f^N(V)\subset V$ for some ...
2
votes
3answers
97 views

Fibonacci Loop Invariants

I've taking an Algorithms course. This is non-graded homework. The concept of loop invariants are new to me and it's taking some time to sink in. This was my first attempt at a proof of correctness ...
2
votes
2answers
113 views

show that the product of two delta functions δ(x)δ(y) is invariant under rotation around the origin.

Show that the product of two delta functions $\delta{(x)}$$\delta{(y)}$ is invariant under rotation around the origin. This is a problem from Zee's textbook on Gravity on page 51. The book was ...
1
vote
1answer
31 views

Given a vector $\vec{p}$ = $<p^1,p^2>$ show $<ap^1,bp^2>$ for $a \ne b$ is not a vector.

In the textbook Einstein Gravity in a nutshell by Zee, on page 43, it claims that given a vector $\vec{p}$ = $<p^1,p^2>$ then for any 2 real numbers $a\ne b$ then the object $<ap^1,bp^2>$ ...
1
vote
1answer
58 views

Scale invariant ODE. Is this general method correct?

Recently, a question I asked had the differential equation $y''=xyy'$. A trick to solving this quickly is to notice that scaling $y$ by $a$ and $x$ by $b$ shows that $a=1/b^2$ is the condition that ...
1
vote
1answer
352 views
2
votes
1answer
20 views

Necessary condition for a finite cyclic sum of length $4$ made of $1$ and $-1$ to be $0$

This is something I observed when I was reading the classic Problem-Solving Strategies by Arthur Engel. I liked the way he solved the following problem: Let $a_1,\ldots,a_n\in\{-1,1\}$ such that ...
0
votes
2answers
41 views

Can we obtain the pair $(1,50)$ with these following operations?

It's a problem from some russian competition: We're given a card with two positive integers $(a,b)$ and we have tree machines which generate another card from the one we insert on it(I assume we ...
0
votes
1answer
34 views

Why is $U$ $T$-invariant?

Let $V$ a finite dimensional vector space and two sub-spaces, $U, W$ such that $V = U \oplus W$. Let's assume $T$ is a linear operator such that $W$ is $T$-invariant. Why is it true that $U$ is also ...
1
vote
2answers
29 views

Application of invariant method

On the board is written the number $18$. Every minute the number is replaced by the product with 2 or 3,or by the quotient of the division with 2 or 3. Show that after $60$ minutes the number cannot ...
1
vote
0answers
62 views

Is there a Fourier invariant basis?

There are some functions which are invariant under Fourier transformation up to scaling factors, eg. sech(pi*x), Gaussian function etc.. Is there a set of basis functions, which form an invariant ...
0
votes
1answer
62 views

Is seven-dimensional cross product rotationally invariant?

For three-dimensional cross product, the following property holds true: \begin{equation} (R\mathbf x) \times (R \mathbf y)=R(\mathbf x \times \mathbf y) \end{equation} where $R\in SO(3)$. Is the ...