# Tagged Questions

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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### 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 ?
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### 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 $\sigma\in S_5$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}_i) = 0$

$(Y_i - \hat{Y}_i)(\hat{Y}_i - \bar{Y}) = 0$ in the image below (third and fourth line of the proof!). Why?
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### 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. ...
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### 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 ...
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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 ...
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### 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 ...
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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 ...
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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-...
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### 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 ...
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### 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 ...
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### 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 $<ap^1,bp^2>$ ...
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 ...