Factorization Definition and 162 Threads

Let a positive definite matrix A be factorized to P and Q, A=P*Q and let an arbitrary matrix B. I am calculating the relative error of the factorization through the norm: \epsilon = \left\| \textbf{A}-\textbf{PQ} \right\| / \left\| \textbf{A} \right\| which gives \epsilon <1\text{e}-16 so I...

Homework Statement Assume n = p_1*p_2*p_3*...*p_r = q_1*q_2*q_3*...*q_s, where the p's and q's are primes. We can assume that none of the p's are equal to any of the q's. Why? Homework Equations The Attempt at a Solution I am completely stuck on this. My understanding of the...

Homework Statement In the expression x2 + kx + 12, k is an integer and k < 0. Which of the following is a possible value of k? (A) –13 (B) –12 (C)  –6 (D)   7 Homework Equations I know it uses the a.c method of factorization but don't know how to use it? The Attempt at a...

Homework Statement Factorize : (x+1) (x+2) (x+3) (x+6)-3 x2 Homework Equations - The Attempt at a Solution Expanding everything , I get x4+12x3+44x2+72x+36 . At this point I tried few guesses using rational roots test. But it appears this has no rational roots. So how should...

Hello, why I can't directly find lim x->3 (x^2+2x-15)/(x^2-5x+6) but I have to factorize them ? Is there any intuitive way to understand that ? Thanks

Let \mathbb{Q}_\mathbb{Z}[x] denote the set of polynomials with rational coefficients and integer constant terms. Prove that the only two units in \mathbb{Q}_\mathbb{Z}[x] are 1 and -1. Help with this exercise would be appreciated. My initial thoughts on this exercise are as follows: 1 and...

Unique Factorization Domain? Nature of Q_Z[x] Let \mathbb{Q}_\mathbb{Z}[x] denote the set of polynomials with rational coefficients and integer constant terms. (a) If p is prime in \mathbb{Z} , prove that the constant polynomial p is irreducible in \mathbb{Q}_\mathbb{Z}[x]. (b) If p and q...

Hi All, I often see this term when factorizing out a matrix from brackets A(some other term)A^T where I assume A A^T represents the square within the bracket term, can someone explain the reasoning behind expressions of this kind or point me in the correct direction Many thanks

Homework Statement Consider the vector a as an n × 1 matrix. A) Write out its reduced QR factorization, showing the matrices \hat{Q} and \hat{R} explicitly. B) What is the solution to the linear least squares problem ax ≃ b where b is a given n-vector. Homework Equations I was...

Homework Statement Claim: If n is a positive integer, the prime factorization of 22n * 3n - 1 includes 11 as one of the prime factors. Homework Equations Factor Theorem: a polynomial f(x) has a factor (x-k) iff f(k)=0.The Attempt at a Solution First, we show that (x-1) is a factor of (xn-1)...

The logic that odd composite with least difference will be factored easily and large difference would factored hardly is wrong. B'coz whatever be the difference between the factors their exist Best Fermat Factors to make the Fermat factorization easier. Please follow the link to know more...

I'm still confused about L U matrix factorization. I'm trying to understand how to do it and why doing so is valuable. Would elementary row operations to solve Ax=b be easier? I'm not in any class. I am looking in the Larson & Edwards Linear Algebra book. Chapter 2. I have trouble...

What is the most motivating way to introduce LU factorization of a matrix? I am looking for an example or explanation which has a real impact.

What is the most motivating way to introduce LU factorization of a matrix? I am looking for an example or explanation which has a real impact.

I am reading Anderson and Feil - A First Course in Abstract Algebra. On page 56 (see attached) ANderson and Feil show that the polynomial f = x^2 + 2 is irreducible in \mathbb{Q} [x] After this they challenge the reader with the following exercise: Show that x^4 + 2 is irreducible in...

Why are QR factorization useful and important?

Why are QR factorization useful and important?

Alright in class, my teacher can factorize quadratics almost instantly. I wanted to know if anyone can tell me how to do it his way... Like if you had 5x^2 + 14x - 3 (x+3)(5x-1) He writes that instantly, I kind of figured out in the first term, you put the sign that the b term...

Homework Statement let A be a UFD and K its field of fractions. and f\in A[x] where f(x)=x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0} is a monic polynomial. Prove that if f has a root \alpha=\frac{c}{d}\in K,K=Frac(A) then in fact \alpha\in A I need some guidance with the proof. Proof...

x^4+1 x^4+2x^2+1-2x^2 (x^2+1)^2-(\sqrt{2}x)^2 (x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1) In particular the second line, seems obvious now that I've seen it but I've never come across in a book before - what is it called?

Homework Statement p(x)=((x−1)^2 −2)^2 +3. From here find the full factorization of p(x) into the product of first order terms and identify all the complex roots. Homework Equations I am having trouble doing this by hand. I know there are four complex roots but can't seem to figure out...

Hi! I read a text were some kind of "Schroedinger-equation" for a neutrino field is being derived. But there is a particular step I do not understand. Consider a Dirac field \psi(t, \vec{x}) of a neutrino, satisfying the Klein-Gordon equation: \left( \partial_{t}^{2} + \vec{k}^{2} +...

Homework Statement I have \frac{5}{4}s_x(f)+s_x(f)cos(2\pi f t_0) + 10 where s_x is 2 between f = -10khz to 10 khz else zero (a rectangle). How do I do spectral factorization when the equation is not in terms of a polynomial of f? All material I can find on this topic have the thing...

Let x,y,z > 0 (x,y,z naturals numbers) Gcd(x,y)=1 Gcd(x,z)=1 Gcd(y,z)=1 z squarefree Factorize x^2 - z*y^2 Thank you.

There is a theorem in algebra, whose name I don't recall, that states that given a polynomial and its roots I can easily factor it so for instance : p(x)=x^2-36 , assuming that p(x) is a real function, p(0)=0 \Leftrightarrow x=6,-6 then p(x) can be written as : P(x)=(x-6)(x+6) I...

Homework Statement Find the QR factorization of A = {1, 1}, {-1, 1} The Attempt at a Solution I just don't know the procedure. I know it means that I need find Q and R such that A=QR, Q be orthogonal, and R be upper triangular. It may be solved by assign Q = {a, b},{c, d}, where ##Q^TQ=1## and...

When I teach GCF to students, I show them how to find via the prime factorization and explain to them how the PF can get you all the factors of a number by multiplying different combinations of the Prime Factors and then proceed to explain why they are supposed to multiply the common Prime...

I'm attempting to write a code for computing the Eigen values of a real symmetric matrix and I'm using the QR algorithm.I'm referring wiki,Numerical Recipees book and other web serach articles. This is a part of the self-study course I'm doing in Linear Algebra to upgrde my skills. My aim...

[b]1. I am asked to write a procedure that will inverse square matices using LU factorization with partial pivoting. [b]2. I am also told that the procedure should return the inverse matrix and report an error if it cannot do so. [b]3. So far I've come up with the code below but...

I encountered a paper in which the authors presented parton-level cross sections as a function of these variables: incoming particle momenta, factorization scale, renormalization scale, and strong coupling constant at the renormalization scale. I used to think that QCD factorization scale should...

Im applying to an REU in San Diego State where the focus will be Nonunique factorization theory but I'm clueless as to what this actually is. Does anybody know anything about this?

Now, for my students and fellow teachers. I am looking to collect a great amount of problems involving factorization, and simplifications of problems. Below is a smal portion of the type of problems I am looking for. "Rules") 1) Simplify a problem, until it can not be simplified...

Homework Statement Suppose you are given the LU factorization for some nxn square matrix A. Assume A is non-singular. This factorization is a result of partial pivoting. Can you use this factorization to solve A^Tx=b for x (given A and b).Homework Equations A^T is the transpose of matrix A...

Hello Everyone, I have a question about LU factorization. I understand that LU factorization provides an upper and lower traingular matrices of matrix A. In matlab, a large matrix was generated, and we plotted the sparsity of A and then the sparsity of L+U and it was less sparse. My...

Homework Statement Let A =[A11 A12; A*12 A22] be Hermitian Positive-definite. Use Cholesky factorizations A11 = L1L*1 A22 = L2L*2 A22-A*12 A-111 A12 = L3L*3 to show the following: ||A22-A*12 A-111 A12||2≤||A||2 Homework Equations The Attempt at a Solution Using the submultiplicative and...

Homework Statement Use Factorization to simplify the given expression. Homework Equations (x^3 + 3x^2 + 3x +1)/(x^4 + x^3 + x + 1) The Attempt at a Solution I can't get to the first step. I forgot how to factor exponents higher than x^2.

I need to factorize large numbers (some of them have about 200 decimal digits). Wolfram alpha is a dead end and programming with python is not working for me too. Any suggestions?

Homework Statement determine whether the following polynomials are irreducible over Q, i)f(x) = x^5+25x^4+15x^2+20 ii)f(x) = x^3+2x^2+3x+5 iii)f(x) = x^3+4x^2+3x+2 iv)f(x) = x^4+x^3+x^2+x+1 Homework Equations The Attempt at a Solution By eisensteins criterion let...

Homework Statement I must understand the following proof. Let A \in \mathbb{R}^{n \times n} be a symmetric and positive definite matrix. Thus there exist a unique factorization of A such that A=LL^t where L is a lower triangular matrix whose diagonal is positive (l_{ii}>0) Demonstration...

The matrix giving the relation between spherical (unit) vectors and cartesian (unit) vectors can be expressed as: \left( \begin{array}{c} \hat{r} \\ \hat{\phi} \\ \hat{\theta} \end{array} \right) = \left( \begin{array}{ccc} \sin\theta \cos\phi & \sin\theta \sin\phi & \cos\theta \\ -\sin\phi &...

Hi I was wondering since i have problems factoring any polynomial past 2nd degree i was wondering if anyone can show a way i can remember for finals ^_^. IE. let's say we have a 3rd degree polynomial. X^3 - 3X^2 +4 i tried looking it up but most don't show how they did the work so i can...

Best fit curve using Q-R Factorization? Homework Statement Homework Equations The Attempt at a Solution So ... It's part (a) that is confusing me. I already factored it into Q and R. But does the Q-R Factorization have to do with best-fit lines? (To be fair, I'm working on homework...

I realize that this might seems to be a strange question, but after doing some coding i realized the following. to brute force the factorization of all numbers less than one million takes around 665 million tests (i.e. does this number divide the original). to do it "smarter" (least i...

So you see it all over the place, \mathbb{Q}(\sqrt{-5}) is not a UFD by finding an element such that it has two distinct prime factorizations...but what about showing that \mathbb{Q}(\sqrt{5}) is a UFD? I'm only concerned with this particular example, I might have questions later on regarding a...

Homework Statement Show that x^2\,+\,x can be factored in two ways in \mathbb{Z}_6[x] as the product of nonconstant polynomials that are not units.Homework Equations Theorem 4.8 Let R be an integral domain. then f(x) is a unit in R[x] if and only if f(x) is a constant polynomial that is a...

I'm confused about how difficult is it to factor numbers. I am reading that it is used in encryption and it is computationally difficult, but it seems to take O(n) from how I see it. For example to factor 6, I would (1) divide by 2 and check if the remainder is 0 (2) divide by 3 and check...

I know that the fundamental theorem of arithmetic states that any integer greater than 1 can be written as an unique prime factorization. I was wondering if there is any concept of negative prime numbers, because any integer greater than 1 or less than -1 should be able to be written as n = p1...

Homework Statement Find all p, prime for which x+2 is a factor of f(x) = 5x4 - 2x3 + 3x2 + 4x - 1 in Zp Homework Equations The Attempt at a Solution So in Zp, x = p-2 I tried the first 4 primes and got the following results: Z3, x=1, f(x) = 9 = 0 Z5, x=3, f(x) = 390 - 1 =/ 0...

For the sake of doing it, I'm trying to factor a quintic polynomial over the reals using a cool technique I found a few days ago. It involves stenciling out the general form of the expression you want and then solving a nonlinear system in which there are more variables than there are...

I'm having a bit of trouble with some ring theory I've been reading about, specifically unique factorization domains. I'm not really clear on how one would go about showing that an element can be factored into irreducibles Homework Statement Let R be an integral domain such that every prime...