## Inverse Polynomial in a ring R November 27    1

I just started working on my Bachelor-Thesis in IT-Security and therefore try to understand the NTRUencryption algorithm. It operates on polynomials in a Ring. My problem is that I don't understand how someone computes the inverse of a polynomial in

## Can a non-square matrix have a full rank November 27    1

Can a non-square matrix have a full rank?I always see cases with square matrix with full rank but seldom with non-square matrix. Can anyone help on this?For example, is the following matrix full rank? A =( 1 3 10) ( 2 3 14) My lecture slide says this

## Skew symmetric $4\times 4$ matrix of full-rank November 27    2

I have come across the fact that a $4\times 4$ skew-symmmetric matrix of full-rank is equivalent to \begin{pmatrix} 0 &\theta_1& 0 &0 \\ -\theta_1& 0 &0 &0 \\ 0& 0&0 & \theta_2 \\ 0& 0& -\theta_2 & 0 \end{pm

## Are positive definite matrices necessarily diagonalizable and when does the famous eigenvalue criterion apply November 27    1

I mean in $\mathbb{C}$ positive definite matrices seem to be self-adjoint. For matrices over real vector spaces this seems to be wrong, but is it still true that they are diagonalizable?Then everyone knows a result similar to this: When a matrix has

## A matrix with purely imaginary eigenvalues is invertible November 27    2

Is it true that if A is a matrix with only eigenvalues that are purely imaginary, then is it invertible?No, it isn't sinceTheorem: Over any field, a square matrix is invertible iff its determinant is not zero iff zero is not one of its eigenvalues.No

## Show: If the adjoint of T is -T, all eigenvalues are purely imaginary November 27    3

Homework question.Let $V$ be a finite dimensional inner-product space over $\mathbb{C}$. Let $T \in L(V,V)$ satisfy $T^*=-T$. Show that all eigenvalues of $T$ are purely imaginary, i.e., if $\lambda$ is an eigenvalue of $T$, then $\lambda = ia$ with

## eigenvalues of the subtraction of a PSD matrix and a rank 1 matrix November 27    1

Let $A$ be a positive semi-definite matrix, and let $C$ be a rank 1 matrix. Prove that $A-C$ has at most one negative eigenvalue.PS: It's easy to show that if $A$ and $C$ commute, then the statement is true, but most of the time they do not commute.S

## Vakil's exercise 5.5.# November 27

I'd like to check that my solution to the following exercise of Vakil's FOAG is correct. I am bothered that we have to use (B) in it, and I want to make sure that I used it in the end for the correct reason.5.5.E. EXERCISE (ASSUMING (A) AND (B)). Sho

## Page 134 and 137 in Hartshorne's Algebraic Geometry November 27    1

In page 134, proposition 6.6 Hartshorne mentions that type 2 is a point $x \in X$ x $\mathbb A^1$ of codimension one, whose image in $X$ is the generic point of $X$. I realized that this point $x$ corresponds to a prime ideal $\mathcal p$ of heigh

## Basic question: Condition for a map associated to a linear series to be an immersion November 27

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without base point. Then $\phi$ is an embedding iff the following

## Gauss curvature using metric and Riemannian curvature November 27    1

I learnt that the Gauss curvature is given by: $$K = \frac {eg - f^2}{EG - F^2}$$where $E, F, G$ are coefficients of the first fundamental form and $e,f, g$ are coefficients of the second fundamental form. However, in a proof that I am reading, I saw

## suggestion for a book on algebraic curves November 27    1

I don't know if i'm asking in the wrong place, but I'm studying the book Algebraic curves of Fulton, and having some problems understanding the final chapters (6,7 and 8). I've found a good help in the Oswaldo Lezama book of Algebraic geometry for th

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