## abstract algebra – \$ g ^ l in langle f_1, …, f_k rangle \$, if the ideal generated by \$ f_1, ~ cdots, f_k, gy -1 \$ in \$ Bbb C[x_1,~cdots,x_n, y]\$ contains \$ 1 \$

To let $$f_1, ~ cdots, f_k$$ Be polynomials in $$Bbb C [x_1, ~ cdots, x_n]$$. I want to show that for a polynomial $$g in Bbb C [x_1, ~ cdots, x_n]$$, $$g ^ l$$ is contained in the ideal that is generated by $$f_1, ~ cdots, f_k$$ for some $$l$$ if the ideal is generated by $$f_1, ~ cdots, f_k, gy -1$$ in the $$Bbb C [x_1, ~ cdots, x_n, y]$$ contains $$1$$.

Because of the term $$gy-1$$I thought this might be similar to Hilbert's zero-digit proof, so I tried to imitate the proof, but I can't see anything. Any clues? Thank you in advance.

## linear algebra – matrix psd inequality, for addition

Given four matrices $$A, widetilde {A}, B, widetilde {B} in mathbb {R} ^ {n times d}$$, if
$$A ^ { top} A approx _ { epsilon} widetilde {A} ^ { top} widetilde {A}$$, $$B ^ { top} B approx. _ { Epsilon} widetilde {B} ^ { top} widetilde {B}$$, do we have
begin {align *} (A + B) ^ { top} cdot (A + B) approx_ {10 epsilon} ( widetilde {A} + widetilde {B}) ^ { top} cdot ( widetilde {A} + widetilde {B})? end {align *}

For a square matrix $$C, widetilde {C}$$, we say $$C approx _ { epsilon} widetilde {C}$$,
begin {align *} (1- epsilon) widetilde {C} leq C preceq (1+ epsilon) widetilde {C} end {align *}

## linear algebra – discrete tomography limited to the interval

There are many methods to find solutions to the equation $$Ax = B$$, Where $$A$$ is $$m times n$$ With $$n> m$$. Are there known methods if the elements of $$x$$ are known to be in the interval $$[0.1]$$?
I am looking for a description or references to such methods.

## linear algebra – upper limit for the condition number of the product of a random sparse matrix and a semi-orthogonal matrix

To let $$G in mathbb {R} ^ {n times m}$$ (m> n, m = O (n)), whose all entries i. distributed as $$mathcal {N} (0, 1) * text {Ber} (p)$$. To let $$V in mathbb {R} ^ {m times n}$$ be a solid semi-orthogonal matrix, d. H. the columns of $$V$$ are orthonormal vectors. Define $$A = GV$$, For what $$p$$ Can we give a polynomial cap for the condition number of? $$A$$ i.e. $$kappa (A) leq text {poly} (n)$$?

Interesting cases / related problems:

1. To let $$V$$ can be defined as $$V_ {i, j} = 1$$ if $$i = j$$ and $$V_ {i, j} = 0$$ Otherwise. To let $$G = (g_1, g_2, ldots, g_m)$$, in this case $$A = GV = (g_1, g_2, ldots, g_n)$$. Hence in this case $$A$$ has the same distribution as $$G$$ except $$m = n$$. This has been investigated by Basak and Rudelson, who have proven this $$kappa (A) leq text {poly} (n)$$ to the $$p = Omega ( log n) / n$$.

2. To the $$p = 1$$, $$G$$ is just a random Gaussian matrix and $$A = GV$$ can also be considered a random Gaussian matrix if Gaussian vectors are isotropic. This is only a sub-case of 1.

## linear algebra – left / right inverse matrix question

For what values ​​of $$a, b, c$$ there is a left and / or right reversal for $$A = begin {bmatrix} 1 & a \ 2 B \ 3 & c end {bmatrix}$$ exist?

We know that a left inverse matrix $$X$$ exists so that $$XA = I_2$$ Where $$I_2$$ is the $$2 times 2$$ So identity matrix $$X$$ is a $$2 times 3$$ Matrix. What do we do next? Thanks a lot.

## ac.commutative algebra – special cases of the embedding problem

Embending problem. To let $$I$$ be the ideal of polynomial algebra $$A = K ^ {[n]}$$, so that $$A / I$$ is also a polynomial algebra with a smaller number $$k$$ of variables. Is it true that $$I$$ is generated by $$n-k$$ Variables of $$A$$.

Definition. Call an ideal of polynomial algebra coordinate-like if it is generated by some coordinates of a polynomial algebra.

I am interested in the following special cases of the embedding problem.

Problem 1. Consider a polynomial algebra $$A = K ^ {[n]}$$ and be $$k$$ Polynomials $$f_1, …, f_k$$. For a polynomial algebra $$B = K [y_1, …, y_ {n + k}]$$ Consider the morphism that sends $$y_i$$ to $$n$$ Coordinates of $$A$$ to the $$i leq n$$ and $$y_j$$ to $$f_ {y-n}$$ to the $$j> n$$. It is obvious that the core of this morphism is the ideal $$I$$ With $$A / I cong B$$. So is it like a coordinate?

Problem 2. If $$I subset J$$ are two coordinate-like ideals of polynomial algebra $$A$$, it is true that $$J / I$$ is the coordinate – like ideal of $$A / I$$?

It is obvious that these problems result from the embedding problem, but are they true?

## linear algebra – the weight of the conjugate partition is greater than the weight of the nullity partition

This question comes from Exercise 4.1 of the lectures on geometric constructions, Kamnitzer-arXiv-Link, and is a consequence of a question that I have asked here that deals with Part 1 of the exercise. This question is part 2.

We get that in this exercise $$X: mathbb {C} ^ N to mathbb {C} ^ N$$ is a not potent matrix with $$X ^ n = 0$$. The partition is connected to it $$mu = ( mu_1, dots, mu_n)$$ With $$mu_i = dim ker (X ^ i) – dim ker (X ^ {i-1}).$$

To $$X$$ We can also map the partition $$nu = ( nu_1, dots, nu_m)$$ where everyone $$nu_i$$ is the size of the $$i$$-th Jordan Block from $$X$$Placing an order to make the 1st Jordan block the largest size, and so on. The young diagram of $$nu$$ has a conjugate partition $$lambda$$where everyone $$lambda_i$$ is the number of $$j$$ so that $$nu_j geq i$$ (i.e. it is the number of Jordan blocks one size larger or equal $$i$$).

The first part shows that for everyone $$k$$, $$mu_1 + dots + mu_k leq lambda_1 + dots + lambda_k.$$ Now I have to show that as $$GL_n$$ Weights, $$lambda geq mu$$.

We have $$lambda – mu = (( lambda_1- mu_1), dots, ( lambda_n- mu_n)),$$ what I want to express as a sum $$k_1 a_1 + dots + k_ {n-1} a_ {n-1},$$ Where $$k_i$$ are not negative integers and $$a_1 = (1, -1.0, points, 0), points, a_ {n-1} = (0, points, 0, 1, -1)$$. In other words, the setup will $$(( lambda_1- mu_1), dots, ( lambda_n- mu_n)) = (k_1, k_2-k_1, dots, k_ {n-1} -k_ {n-2}, -k_ { n-1}).$$

Starting from the left, we get an inductive equation $$k_i = ( lambda_1 + dots + lambda_i) – ( mu_1 + dots + mu_i),$$ but I can't show that $$– k_ {n-1} = lambda_n- mu_n tag {*}.$$ I'm pretty sure my calculations are correct, and the only place we use the part 1 inequality is to show that the constants $$k_i$$ are not negative, but that's all right – the only part I'm sticking to is showing (*).

## linear algebra – canonical form and basis of the orthogonal operator

Find the canonical form and the canonical basis of the orthogonal operator $$f$$ which has the following matrix in an orthonormal basis $$A_f = frac {1} {3} begin {bmatrix} 2 & -1 & 2 \ 2 & 2 & -1 \ -1 & 2 & 2 end {bmatrix}.$$

Approach: We know that there is a canonical basis for every orthogonal operator, so the matrix of the operator $$f$$ is on that basis $$begin {bmatrix} pm 1 & 0 & 0 \ 0 & cos varphi & – sin varphi \ 0 & sin varphi & cos varphi end {bmatrix}.$$ Since the determinant and the trace of the matrix of the linear operator are the same in every basis, we make the following remark: since $$det A_f = 1$$ then the first element of the first line should be the same in canonical form $$1$$. Since $$text {tr} A_f = 2$$ then $$2 cos varphi + 1 = 2 Leftrightarrow cos varphi = frac {1} {2}$$. In order to $$sin varphi = pm dfrac { sqrt {3}} {2}$$.

It also follows from this $$1$$ is an eigenvalue of the operator $$f$$ and is the corresponding eigenvector $$e_1 = frac {1} { sqrt {3}} (1,1,1)$$. In order to $$e_1$$ can be taken as the first vector of the canonical base and we know that the canonical form is $$begin {bmatrix} 100 \ 0 & cos varphi & – sin varphi \ 0 & sin varphi & cos varphi end {bmatrix}.$$

I cannot consistently solve the following questions myself:

1) How do you find the rest of two canonical vectors?

2) And what value of $$sin varphi = pm dfrac { sqrt {3}} {2}$$ I have to take?

## Representation theory – algebras based on Frobenius algebra

To let $$A$$ be a commutative Frobenius algebra over a field $$K$$ (We can accept that $$A$$ is local).

To let $$B = {v_i }$$ be a vector space basis of $$A$$ with the unity of $$A$$.
To let $$M_i: = v_i A$$ and $$M: = bigoplus _ {} ^ {} {M_i}$$ and $$C: = underline {End_A} (M)$$ the stable endomorphism ring of $$M$$.

The question is $$C$$ regardless of the base chosen $$B$$ up to isomorphism?

## linear algebra – multi-variant Gaussian

I have a few questions about the multivariate Gaussian formulation. I've seen a lot of videos and read Wikipedia, but I don't quite understand why things happen.

In this picture we have the general form of a multi-variate distribution.

My first question is roughly $$Sigma$$ if it is the correlation or covariance matrix. I've seen different sources related to both, and I don't know why because they are different.

My second question concerns the form $$x ^ T Sigma x$$ I do not understand this general form of wrapping a matrix in two x. WHY is that because we want x ^ 2? and if that's the case, why can't we do that? $$Sigma x ^ Tx$$? Finally, that $$| Sigma |$$ is the determination of $$Sigma$$ but in the univariate case it is the standard deviation determined by $$Sigma$$ equal to the standard deviation?

Any help on this intuition would help me a lot, thanks.