# 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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### 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 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 $k = 1,2, \ldots ,\text{dim}V$ ...
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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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### 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 .
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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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### 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 ...
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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. ...
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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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### 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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### 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 ...