4
votes
1answer
28 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
0
votes
0answers
56 views

Transform one curve into another

I have been working on something for a while now, and I can't really get my head around it. I consider two curves with data points and want to determine the most optimal transform from one to another. ...
3
votes
2answers
84 views

Manipulating identities

I'm having some trouble deriving certain identities. If $$S(z) = \prod_{i=1}^n (z-z_i)$$ then how can I write $$\frac{1}{S(z)}\frac{d^2S}{dz^2} = \sum_{i=1}^n\frac{1}{z-z_i}\sum_{j\neq ...
0
votes
1answer
27 views

Matrix Vector Multiplications

If your coordinate system is assumed to be right-handed and given in the following Orthogonal Matrix. M =[l1 m1 n1;l2 m2 n2;l3 m3 n3] suppose we multiple this ...
1
vote
1answer
46 views

Computing the intersections of arbitrary number of planes

Imagine three sets of planes (A, B, and C). The planes in each set are parallel and evenly spaced and so can be defined by a plane equation and a scalar representing the distance between planes. If ...
3
votes
1answer
22 views

Generic points as coefficients of polynomial kernels?

I am reading the paper Dual-to-Kernel Learning with Ideals. Here is part of it: The definition/motivation of genericity in Wikipedia are A generic point of the topological space $X$ is a point ...
1
vote
2answers
65 views

Does congruence guarantee length conversion?

Suppose that a linear transformation $M:R^2 \rightarrow R^2$ maps a triangle $ABC$ to a congruent triangle $A'B'C'$ ($\{A, B, O\}, \{B, C, O\},\{C, A, O\}$ are not colinear, and $A,B,C\neq O$) Is it ...
0
votes
3answers
32 views

Proof of normalvector on a plane

I found that, for the plane with linear equation: Ax + By + Cz = 0, that the vector a with coordinates: (A, B, C), is a normal vector on that plane. Where does that come from? And can someone provide ...
0
votes
2answers
43 views

Distorted Unitary matrices

Let $U$ be an unitary and $D$ be a diagonal matrix. We know that for all vectors $v$ on the sphere $Uv$ is on the sphere and, $$\langle Uv,Uv\rangle=\langle v,v\rangle.$$ What are the vectors $v$ on ...
0
votes
0answers
31 views

Vectors on a Sphere

Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be unitary matrix. Let $r\in\Bbb Z_+$ be a fixed integer. $(1)$ For a vector $v$ ...
5
votes
2answers
237 views

Is evaluation homomorphism surjective?

Let $A^n$ be an affine space over $\mathbb{C}$ and let $\mathbb{C}[X_1,\cdots,X_n]$ be the polynomial ring of $n$ variables. Then $A^n\to (\mathbb{C}[X_1,\cdots,X_n])^*$ by evaluation homomorphism, ...
1
vote
0answers
27 views

Is Grassmann-Plucker relation implied by 3-term Grassmann-Plucker relation?

It's a problem from book "Oriented Matroid", problem 3.16. More exactly: For $n$ elements $i_1,\dots,i_n$ and an anti-symmetric function $\det$ on${\{i_1,\dots,i_n\}}^r$. we have: ...
0
votes
0answers
37 views

Relations between minors of a matrix.

motivation: I'm looking at the Segre embedding, given by (for this example) $\mathbb{P}(U) \times \mathbb{P}(V) \rightarrow \mathbb{P}(U\otimes V)$, $([u],[v]) \mapsto [u\otimes v]$. This is an ...
2
votes
3answers
106 views

Find radius of a circle which is tangent to three known lines

I need to find the equation for a circle which is tangent to the following three lines: y=0 x=0 y=-x+0.338334 For the last tangent line equation, I know that it is tangent at the point (0.169167, ...
8
votes
1answer
132 views

Linear combination of matrices

Let $A, B, C, D$ be four linearly independent symmetric 3 x 3-matrices over $\mathbb K$. Show that some linear combination of these matrices is a matrix of rank 1. I know it is supposed to be a ...
1
vote
0answers
52 views

Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial $$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n} $$ After the linear change of ...
2
votes
1answer
63 views

Projective equivalence

Definition Two projective plane curves $F$ and $G$ are projectively equivalent if there is a $\varphi_A\in PGL_2(k)$ such that $F(x,y,z)=G(a_{00}x+a_{01}y+a_{02}z,a_{10}x+a_{11}y+a_{12}z, ...
2
votes
0answers
26 views

Why is the definition of intersection multiplicity independent of choice of homogeneous coordinates?

I am using the following definition of intersection multiplicity of two algebraic curves $\mathcal C, \mathcal D$ of degrees $m,d$ without common components in $\Bbb P^2(\Bbb K)$: Let $P \in \mathcal ...
1
vote
1answer
34 views

Why $d_xG_{|T_xX}=0\implies d_xG=\lambda_1d_xF_1+\cdots+\lambda_md_xF_m$?

I'm trying to understand this proof of this theorem of a book I'm reading in basic algebraic geometry: Theorem Let $X$ be a closed affine subset and $x\in X$. The restriction of $d_x$ to $I_X(x)$ ...
-1
votes
1answer
58 views

find a common plane which contains two points (NP hard?)

In this problem the coordinates of 4 points are given. $p_{0}=(x_{0},y_{0},z_{0})$, $p_{1}=(x_{1},y_{1},z_{1})$, $p_{2}=(x_{2},y_{2},z_{2})$ and $p_{3}=(x_{3},y_{3},z_{3})$ I need to find the ...
2
votes
2answers
313 views

Decomposing an Affine transformation

An affine transformation is composed of rotations, translations, scaling and shearing. In 2D, such a transformation can be represented using an augmented matrix by $$ \begin{bmatrix} \vec{y} \\ 1 ...
3
votes
3answers
82 views

What is the dimension of this Grassmannian?

Why is $2\times 3$ the dimension of $Gr_2(\mathbb{R}^5)$? and can one use the dimensions of Lie groups to derive this dimension? Note: $Gr_2(\mathbb{R}^5)$ denotes the Grassmannian of all ...
0
votes
1answer
51 views

What is this dimension?

What is the real dimension of the cone of $2$ by $2$ Hermitain matrices with at lease one eigenvalue that is $0$?
37
votes
4answers
1k views

How to identify surfaces of revolution

Given a surface $f(x,y,z)=0$, how would you determine whether or not it's a surface of revolution, and find the axis of rotation? The special case where $f$ is a polynomial is also of interest. A ...
0
votes
1answer
65 views

what are the various fields in which circle is treated as infinite sided regular polygon?

What are the various fields in which circle is treated as infinite sided regular polygon? What I actually mean is , "can u suggest me some applications where circle is treated as infinite sided ...
2
votes
1answer
98 views

How to extract the indeterminates from a set of polynomial?

I am a biologist and I am facing a huge problem. I would like to extract the indeterminates of a set of polynomials, for example, I have: $f_{1} = \{\\x_{3}^{2} + x_{1}*x_{2} + x_{1} + x_{1}*x_{3},\\ ...
0
votes
1answer
212 views

How to find whether the line is inside the polygon or outside.

I have a polygon How can i prove whether the black color line lies outside the polygon or inside the polygon . Given the coordinates of the black line and all the vertices of the polygon.
2
votes
1answer
212 views

Rotation matrix convention; successive rotations in intermediate coordinate systems or not

I am getting very confused about the different conventions used for rotation matrices. Thing is I want to accomplish (3) successive rotations each time in the newly defined coordinate system. I use ...
1
vote
1answer
61 views

How do i define a plane orthogonal to a given one?

I have a plane $P_1$ given by the equation: $$ ax+by+cz+d=0$$ I want to find an orthogonal plane to that. I know that their normal vectors should be orthogonal so the normal vector $v_1$ of $P_1$ is ...
4
votes
0answers
86 views

What does “generic” mean in this context, and is it related to generic points in algebraic geometry?

In "The characteristic polynomial and determinant are not ad hoc constructions" by Skip Garibaldi, available at http://arxiv.org/abs/math/0203276 the characteristic polynomial is defined as the ...
0
votes
0answers
30 views

What can I say about these two points… [duplicate]

Two points on the graph of $y=kx^p$ are labeled $A$ and $C$. Point $A$ has coordinates $(a,b)$, where $0<a<1$ and point $C$ has coordinates $(c,d)$, where $1<c$. If we are told that that ...
1
vote
1answer
34 views

How to properly sort a set of axis-aligned boxes so they are drawn correctly under this projection?

Given a set S of axis-aligned, non-overlapping boxes {x,y,z,w,h,l}, where x,y,z are their center-positions and w,h,l their width, height and lengths, and given the following orthographic projection: ...
2
votes
2answers
56 views

Quadratic Bezier curves representation as implicit quadratic equation

A quadratic bezier curve from points P1=(x1, y1) to P3=(x3, y3) with control point ...
1
vote
1answer
47 views

Proof convex polyhedron with line does not contain a corner if closed

The excercise I am struggling with is the following: Given a convex closed polyhedron that contains a line, the question is, whether this polyhedron can also contain a corner. My idea was to make a ...
1
vote
0answers
41 views

Edge in a convex polytope

I want to show that a convex polytope $A$ that is an intersection of half-spaces contains an edge if $ A=\{x \in \mathbb{R}^n|Ax=0 \wedge x \ge 0\}$, where x greater equal 0 means, that all components ...
1
vote
0answers
74 views

Classification of Cones

I am attempting to classify (convex rational) cones in $\Bbb{R}^2$. We say here that $\sigma\subset\Bbb{R}^2$ is a cone if there exist $u,v\in\Bbb{Z}^2$ which $\Bbb{R}$-span the whole plane ...
0
votes
1answer
117 views

Points from an affine subspace with equal distance from given points

Given vector space $\mathbb{R}^3$ with dot product defined as $x \cdot y = 2x_1y_1 + 3x_2y_2 + x_3y_3$ where $x = (x_1,x_2,x_3),y = (y_1,y_2,y_3)$ and given an affine subspace $W: x - y - z - 2 = 0$ . ...
2
votes
1answer
69 views

Polynomial Equations for the Rank of a Power of a Matrix

If I have some $n \times n$ matrix $X$ (in my case, I happen to know that X is nilpotent and in Jordan normal form), how can I write the condition that $\text{rank}(X^r)= k$ as a polynomial equation ...
4
votes
2answers
206 views

Arrangements of affine hyperplanes

Fix $n>0$ and $X\subseteq\mathbb{R}^n$. Call a function $f:X\longrightarrow \mathbb{R}$ linear if it is of the form $$ f(\bar{x})=a_1x_1+\ldots+a_nx_n+b $$ for some $a_i,b\in\mathbb{R}$. Now ...
3
votes
1answer
87 views

Toric Varieties from Cones

Consider the lattice $N=\Bbb{Z}^d$ spanned by $e_1,\dots,e_d$ and the cone $$\sigma=\text{Cone}\{e_1,\dots,e_k\}, \quad k<d.$$ I am trying to understand why the toric variety $V_\sigma$ obtained is ...
2
votes
1answer
85 views

Each affine transformation $\mathbb{R}^n \rightarrow \mathbb{R}^n$, wose derivative doesn't have eigenvalue 1, has a fixed point

Please help me prove, that each affine transformation $\mathbb{R}^n \rightarrow \mathbb{R}^n$, whose derivative (the linear transformation connected with this affine transformation) doesn't have ...
7
votes
1answer
88 views

Finiteness of groups preserving a symmetric positive definite bilinear form

This question arises from reading the note Hodge cycles on abelian varieties by P. Deligne (notes by J.S. Milne). Suppose we are given a group $G$ (for example, either a fundamental group $\pi_1(S, ...
4
votes
1answer
117 views

Every conic in $\Bbb{P}^2$ equivalent to $XZ - Y^2$ - what is meant by hint here?

I am looking at Miles Reid's UAG book. There he claims that every projective conic is projectively equivalent to $XZ = Y^2$. He asks to show that $Q$ a non-degenerate quadratic form is such that ...
3
votes
2answers
68 views

Formalisation of equivalence of “vectors” and “points”

In all the linear algebra courses I've taken, the notion of a "geometric vector" has not been rigorously defined. All that has been explained is that we can think of them as algebraically equivalent ...
0
votes
1answer
1k views

Converting parametric equation to implicit form

So I have the equation defined in homogeneous coordinates $[w; x, y]$ as $[1+t^2; 1-t^2, 2t]$ $$w = 1+t^2$$ $$x = 1-t^2$$ $$y = 2t$$ If I do $w+x-y$ I get $-2t+2$, so $t = -(w+x-y-2)/2$. I was then ...
0
votes
3answers
141 views

Equation of a line in homogenous coordinates given 2 points in affine coordinates

So if I have 2 points $A$ and $B$ such that $F(A) = (1; a, a^3)$, and $F(B) = (1; b, b^3)$. how do I find the equation of this line in homogeneous coordinates? So I know how to get a line the ...
12
votes
1answer
244 views

Variety of pairs of product-zero matrices

Here's an old qualifying exam question I got stuck on. Consider the variety $X$ of pairs of matrices $(A,B)$ satisfying $AB = BA = 0$ (with entries in some field). What are the irreducible components ...
1
vote
2answers
46 views

Find $A^{-1}$(W) of linear manifold W

Given linear map $A:\mathbb{R}^2\to \mathbb{R}^4$ defined as $$A = \begin{pmatrix} 1 & 1 \\ 1 & -1 \\ 0 & 2 \\ 3 & 1 \end{pmatrix}$$ and linear manifold $ W \subset ...
1
vote
1answer
71 views

Projective Equivalence of $(n+2)$-tuples in $\mathbb{P}^n$

Let $p_1,...,p_{n+2}$ and $q_1,...,q_{n+2}$ be two sets of distinct points in general position in $\Bbb P^n$ (there may however be overlaps between the two sets.). Then there exists $\phi\in ...
2
votes
1answer
179 views

Expectation of a multivariate Gaussian over a plane

For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation : $E[X|X^Tb = c]$ ...