Decomposition Definition and 414 Threads

What is the best way of introducing singular value decomposition (SVD) on a linear algebra course? Why is it so important? Are there any applications which have a real impact?

This is purely conceptual and I'm just looking for opinions on whether its misguided or, indeed, plausible. From what I understand about Fourier decomposition we can break down an analog signal into component sinusoidal waves. My thinking is that the sound system at a nightclub can be...

I'm studying the decay K->ππ and I have some doubts on the isospin decomposition. We know that the state (\pi\pi) can have total isospin 0 or 2. Now, if we remember that in the isospin representation we have |\pi^+\langle=|1,1\langle, |\pi^0\rangle=|1,0\rangle and |\pi^-\rangle=|1,-1\rangle...

hi everybody Today I have a question about Kronecker products, If you have a direct answer it is perfect but if not, any kind of paper reference might work as well. now say I have to matrices A and B in general there is nothing special about them. They are not hermitian or triangular or what...

So I'm trying to figure out how to decompose the following using octave: 85000/[(s^2+250^2)(0.2s^2+40s+10000)] I tried using the residue command but I think that only works if the polynomials have real roots, which these don't. When I do use residue I get the following: b =...

A. Given the exact locations of N electrons - can we find out the total wave function and the individual wave functions by some law of nature? B. Given the exact total wave function - can we find out the coordinates of each electron and the indiviual wave function of each electron by some law...

I am just into reviewing abstract algebra and came across a theorem I'd forgotten: http://en.wikipedia.org/wiki/Finitely-generated_abelian_group#Primary_decomposition (I linked to the theorem instead of writing it here just because I'm not sure how to write all those symbols here) Anyway...

Homework Statement I have J=\begin{bmatrix} \frac{\pi}{2}&0&0\\ 1&\frac{\pi}{2}&0\\ 0&1&\frac{\pi}{2}\\ \end{bmatrix} I need to find \sin(J) \text{ and } \cos(J) \text{ and show that } \sin^{2}(J)+\cos^{2}(J)=I Homework Equations The Attempt at a Solution I have the...

Homework Statement A is arbitrary linear map of the complex vector space s.t. A: V-->V. 1) Show that there exist unitary matrices s.t A=V*DU where D is diagonal and its entires are non-negative. 2) Show that part 1 holds if and only if there exist orthonormal bases {u_i} and {v_i} s.t...

Dear all, Homework Statement For practice, I'm trying to solve an exercise I've found on the Internet. A screen shot of it is attached. I would like to be sure about my approach, especially when it comes to question 2. Let's say, for the sake of simplification, that the reaction is as...

Homework Statement In a first order decomposition in which the rate constant is 0.03 sec-1, how much of the compound (in mol/L) is left after 39 sec, if there was 2.00 mol/L at the start? I'm using a few equations and trying to plug it in but I don't know whether they are appropriate or not...

I'm working on the hydrothermal decomposition of hemicellulose and one of the reaction products is acetic acid. The incomplete reaction model I'm using says 20% of my original hemicellulose is converted into acetic acid. In comparison to other papers this seems to be too high. So were does my...

Theorem: Prove that there exist $n$ edge disjoint Hamiltonian cycles in the complete graph $K_{2n+1}$. ---------------------------------------------------------------------------------- I have found two constructive proofs of this over the internet. But I would like to prove it...

I have been having trouble of late with partial fraction decomposition. Not so much the maths, but the intuition behind it. What I mean by this, but a question in front of me, I now what procedure to follow to get the answer, but I don't get why you follow the said produced. I will give an...

Homework Statement Decompose the following matrix using QR decomposition \begin{bmatrix} 4 & 1 \\ 3 & -1 \end{bmatrix} the answer is \begin{bmatrix} .8 & .6 \\ .6 & .8 \end{bmatrix} The following matrix is supposed to be next to the previous but I can't figure out how to do that. Any...

Hey guys! I'm new here, so forgive me if I'm posting in the wrong section. I recently picked up a book on robotics and it had a section about rotation matrices. I'm having a difficult time with the decomposition of rotation matrices. Everywhere I look, I can find the the equations for the roll...

I have a puzzling question that a gentleman discussed and have differnt view on the problem, the probelm goes as follows: What is know is the Resultant of a Vector Addition, the goal is to find the two vectors from which the resultant came from. The other piece of information that is know is...

Hi, Consider P = \boldsymbol{\nabla} f +\boldsymbol{\nabla}\times \bold{A} where f and A are scalar and vector potentials, respectively, and P is strictly positive and well behaved, and only nonzero in a domain \mathcal{D}. I want to find how the magnitude of \int \boldsymbol{\nabla} f dV...

Homework Statement If g(x) = 2x + 1 and h(x) = 4x^2 +4x + 7 , find a function f such that f o g = h Homework Equations The Attempt at a Solution Well, I know I have to decompose h into two parts: the give one being g, and the other f. But I can't seem to perceive any relation between g and h...

Homework Statement Decompose \mathbb{C}^{5}, the 5 dimensional complex Euclidean space) into invariant subspaces irreducible with respect to the group C_{5} \cong \mathbb{Z}_{5} of cyclic permutations of the basis vectors e_{1} through e_{5}. Hint: The group is Abelian, so all the irreps...

Homework Statement For a system A consists of two parts A' and A'' which interact only weakly with each other, if the states of A' and A'' are labeled respectively by r and s, then a state of A can be specified by the pair of numbers r,s and its corresponding energy E_{rs} is simply...

As part of a project I have been working on I fin it necessary to manipulate the following expression. e^(icx)/(x^2 + a^2)^2 for a,c > 0 I would like to decomp it into the form A/(x^2 + a^2) + B/(x^2 + a^2) = e^(icx)/(x^2 + a^2)^2 but I am having trouble getting a usable outcome.

I'm have a little trouble understanding PA=LU, I have no problems with A=LU but can't seem to figure out the Permutation matrix. So I have summarised the process I am using let me know where it can be improved. Step 1: Using Gaussian Elimination with partial pivoting reduce A to form a...

I apologize for not having any attempted work, but I have no idea how to even begin tackling this proof. Any direction would be greatly appreciated! Mike Homework Statement Let V be a vector space, Let W1, ..., Wk be subspaces of V, and, Let Vj = W1 + ... + Wj-1 + Wj+1 + ...

I have a large sparse symmetric matrix and I'd like to know the number of its negative eigenvalues. To this end, I should perform an LDLT decomposition of the matrix and count the number of negative diagonal entries of the D matrix. This would be equal to the number of negative eigenvalues...

\begin{bmatrix} 1 & 2 \\ 2 & 1 \\ \end{bmatrix} = A The eigenvalues of A are 3 and -1. The eigenvectors are (1,1) and (-1,1), respectively. I'm not sure how to proceed.

Homework Statement \frac{2e^3}{((s^2)-6s+9)*s^3} you can factorize the denominator into s,s,s,(s-3),(s-3) that gives you 5 residuals. the first 3 should all be the same value but that's apparently not correct, so where am I going wrong?

Homework Statement I'm supposed to decompose 1 / x(x2 + 1)2 Also, we haven't learned matrices yet so I can't use that technique to solve it. Homework Equations None. The Attempt at a Solution 1 / x(x2 + 1)2 = A/x + (Bx + C) / (x2 + 1) + (Dx + E) / (x2 + 1)2 I multiplied...

Homework Statement Okay so I'm supposed to find the least squares solution of a set of equations, which I can do, but it adds that I must use QR decomposition. I don't really know how to apply QR decomposition to this problem. Problem: Find the least squares solution of x_1 + x_2 = 4...

Homework Statement Let S and T be nonempty sets of real numbers such that every real number is in S or T and if s \in S and t \in T, then s < t. Prove that there is a unique real number β such that every real number less than β is in S and every real number greater than β is in T. The Attempt...

I have been doing some research on ozone recently, and I have a question that none of my resources have answered... The following reaction is an exothermic reaction, but how would I calculate the amount of heat energy is released? O_{3} \underbrace{\rightarrow}_{uv\ light} O_{2} + O...

Actually I am new to this topic. I read few tutorials about LU decomposition of matrix in the net. A = LU ; A - actual matrix, L - Lower triangular matrix, U - Upper triangular matrix.Few people say that, principal diagonal elements of L should be unity. Some others say that, principal...

Assuming the matrix is positive definite (necessary for cholesky decomposition). Which is faster? Which is more accurate? Is there a reliable source that has all the most common decompositions listed in order of accuracy and speed?

I've been trying to invert a real symmetric matrix and the inverse that I compute via eigenvalue decomposition is not the inverse (using QV^-1Q^T), the stranger thing is that QVQ^T gets back my orginal matrix matrix. Even more unusual is that the matrix starts off at approximately identity (in...

As we know the algebra of SU(3) consist of two Cartan generators and 6 raising and lowering operators. We define the eigenstates of the Cartan operators as u,d,s, correspoding to the three lightest quarks. Now when we study the 3\otimes 3 tensor product we can show that the Hilbert space of...

Let F= R or C, and A = [1 2 3] is considered as linear operator in F3 [0 1 2] [0 0 1] then the minimal polynomial of A = (x-1)^3, can we say that the primary decomposition thm doesn't give any decomposition, can we find an invertible P s.t P^-1*A*p is a block diagonal matrix?

Homework Statement Show that n/(n+1)!=(1/n)-(1/(n+1)!) I am totally lost on the algebraic steps taken to come to this conclusion. It is for an Infinite series. Thanks

Homework Statement I have a Stochastic Processes test coming up soon and we are taught about the first passage decomposition. I understand this but it then says that it changes it to a generating function Homework Equations Generating function Pij(s) = Ʃ pij(n)sn...

Recently a student brought the following to my attention from Weinberg’s Quantum Theory of Fields, Volume I, from page 177, which I must admit that it stumped me. Here Weinberg introduces the concept of Cluster Decomposition: “It is one of the fundamental principles of physics (indeed, of all...

I want to solve the linear equation below: Ax = b For this purpose, I'm writing a C++ code. I have written both routines for decomposing A matrix to L and U matrices, and for calculating inverse of A matrix. I may multiply both sides with A-1: Ax = b A^{-1}Ax = A^{-1}b x =...

I can not find any useful online tool that solves you a system of equations using Gaussian elimination and LU decomposition. So just a system like: -3X+4X+9X+4x=-2 9X+2X+1X-5X=2 etc. Just about 4 lines. several lines. So basically an online tool where you can just plug in the...

I'm trying to read this proof, and I'm stuck on the inequality on page 27 following the statement "It follows that every measurable subset..." Why does it hold? The theorem is about signed measures, i.e. functions that are like measures, but can assign both positive and negative "sizes" to...

Yes, another of these. How do you decompose (z3+1)/(z(1-z)2) ? I've tried A/z + B/(1-z) + (Cz+D)/(1-z)2 and a slew of others that don't work. Thanks a bunch!

Homework Statement Hi If we have a matrix M, we can always make a singular value decomposition. If the matrix has full column rank (= is invertible), then the singular values are all nonzero, otherwise they are not all nonzero. Now, we can also associate a condition number to a matrix...

A = PD[P][/-1]; A: square matrix; D: is a matrix of Jordan canonical form P:is EigenVectors..(p1,p2,p3,p4...pr) Is it possible to permute the sequence p1,p2,p3,p4...pr into other form? I know it should be possible to permute... How should permute it and what significance it has? Why is...

The plane wave decomposition is mathematically universal? 1. My questions is: Can "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" be represented as a sum of uniform plane waves? Note: r=0 is a singularity. This is actually the potential...

Say I have three elements: A, B, C. I can list all the permutations by going alphabetical in the first element, then the second, then the third, and so on, like so: ABC ACB BAC BCA CAB CBA What I'm wondering, is given a number N, how do I decompose this into knowing what permutation it...

I took an Intermediate Linear Algebra course all last year (two semesters worth) and we covered the CDT. My professor didn't teach it well, and I got my first B- in university because of it (didn't affect my GPA but still irritating). I didn't understand a lot of the canonical form stuff...

Hi, The vector potential for elctrodynamics, A_{\mu}, can be decomposed A_{\mu}\in\mathbf{0}\oplus\mathbf{1} but the \mathbf{0} part we ignore. My question is: do we ignore the scalar part because of experiment, or is it ignored for mathematical reasons I am naive about? Thanks,

Homework Statement Find the projectors of matrix A (I am not sure how to write it in matrix form here so I have written out each row of the matrix) Row 1 : [ 2 1 1] Row 2: [ 2 3 2] Row 3: [ 1 1 2] Homework Equations A theorem in my textbook says the Lagrange...