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Yes there are infinitely many such values of $n$ and the sequence satisfies the rule you observed. The proof is straightforward but technical. Let $x_1,\dots,x_n$ be the solution. Add $x_0=0$ and $x_ …

If you are looking for a geometric classification of all affine transformations of the Euclidean plane (not to mention higher dimensions and other fields), there is probably no such thing. It is possi …

A stronger fact holds: If $\lambda_1\le\lambda_2\le\dots\le\lambda_m$ are the eigenvalues of $A$, and $\mu_1\le\mu_2\le\dots\le\mu_p$ are eigenvalues of $Z^TAZ$, then $\mu_k\ge\lambda_k$ for all $k=1, …

This is true and well known. By the minimax principle, $\alpha_k$ is the minimum over all $k$-dimensional subspaces of the norm of the quadratic form $v\mapsto(v,Yv)$ restricted to the subspace. And s …

I think it is best to settle this problem geometrically, that is if you think of matrices as linear maps from $\mathbb R^m$ to $\mathbb R^n$. The images of these maps are $r$-dimensional linear subspa …

A quick counter-example to the question as stated is $a_0=0$, $a_1=1$ $a_2=-1$. Since $2arccos(0)-arccos(b)-arccos(-b)=0$ for all $b$, we have $2M_1-M_2-M_3=0$ where $M_1, M_2, M_3$ are the rows of th …

A homogeneous linear system does not have a nontrivial nonnegative solution if (and only if) some linear combination of the equaltions yields a nontrivial equation with nonnegative coefficients. Nothi …

The image of the $L_1$-ball is the convex hull of the images $f(\pm e_i)$ of the basis vectors $e_1,\dots,e_n$ and their opposite ones. So you are given $n$ pairs of opposite points in $\mathbb R^m$ a …

I don't know what was meant in that exercise, but your revised conjecture is certainly true and well-known. Here is an elementary proof. The assumptions (local injectivity, continuity and segment-to-s …

The catch is in the word "canonical". If $V$ is positive, then $V^\perp$ is transverse to $V$ and hence naturally isomorphic to $L/V$ (by means of the projection $L\to L/V$ restricted to $V^\perp$). …

The sum is zero for all $k<p^2-1$. Assume that $k$ is a multiple of $p-1$ and $k<p^2-1$. Divide all matrices into classes of the form $\{A,A+1,A+2,\dots,A+(p-1)\}$ . Summimg the $k$th powers over su …

No, being large component-wise is not enough. Consider the system $$ \begin{cases} 2x+y+z = b_1 \\ x+2y+z=b_2 \end{cases} $$ If $x,y,z\ge 0$, then obviously $b_2\le 2b_1$. So, for $b=(N,3N)$ there are …

No. Consider solid angles in $\mathbb R^2$. They all (except the half-plane) are isomorphic, yet one may be strictly included in another. Furthermore, a generic cone in $\mathbb R^d$, $d\ge 3$, has a …

To me continuity is more geometric and intuitive than the rest of the argument (which is purely algebraic manipulation). So I take the liberty to mis-read you question as follows: Is it possible to …

I believe that the statement you want is not true. In $X=\mathbb R^3$, begin with the standard cone $x^2+y^2<z^2$ and perturb it so that the resulting cone $K$ is symmetric to its Euclidean dual throu …