Questions tagged [invariance]

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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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. Prove that there is no way to get the value 100,…
Analysis15
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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 $C^2$ into itself.) Besides the usual, mechanical…
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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 give a doughnut to Alice at any time. 2) If…
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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 both change into green chameleons). When two of…
shoestringfries
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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. Re-reading my algebra text, it dawned on me that the reason…
Tac-Tics
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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 $M^3 = 0$), not diagonalizable, and not invertible. Basically, I need to show…
user276487
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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 don't lose the first one, when we put one card on it…
onlyme
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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 $T$-invariant?
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Proofs involving Subspace Invariance

Let $ V $ be a finite dimensional vector space over a Field $ F $ and let $ T \in End(V) $. i) If $ S \in End(V) $ is such that $ ST = TS $, show that $ Im(S) $ is a $T$-Invariant Subspace of $ V $ My approach involved showing that $Im(S)$ is a…
Gregory Peck
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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 $\lambda\in\Bbb R\setminus\{0\}$? The rank and the…
Joe
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Is this proposition posible?

In a board, you have $13$ White round pieces, $15$ Black round pieces, and $17$ Red round pieces. In each round you can choose two different color pieces and change them with two other pieces of the remaining color. Doing this every round, it's…
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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 $V$ matrices for the action on $W$ wrt basis…
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Which blocks of $5×5$ grid can have the last lit lamp?

There is one off lamp in each block of a $5×5$ grid. Each round we can choose a block and change the status of the lamp inside it. As a result, lamps in the neighboring blocks (sharing a common side) also change their status i.e., from on to off or…
Dojou
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Prove that there are 2 numbers whose difference is divisible by 2n.

I have been trying to solve this problem using pigeon hole principle but I think it has some subtleties I might not be paying attention to : We have the natural numbers $1,2,...,2n$ and we have written them arbitrarily in $2n$ numbered places. If…
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