Decomposition Definition and 414 Threads

I have to show that a generic vector can be decomposed into an irrotational and solenoidal component: V(r) = -Grad[phi(r)] + Curl[A(r)] I'm not sure how to start. Do I need to take the curl or div of V and use a vector identity? Any help would be greatly appreciated!

For the reaction at 523K PCl5 (g) <-> PCl3 (g) + Cl2 (g) and Kp = 0.500 What percentage of PCl5(g) will decompose if 0.100 atm of PCl5(g) is placed in a closed vessel at 523K? I am unsure of how to approach this problem, if anyone could give some insight I would greatly appreciate it.

Hi everyone, I have a problem with the following matrix equation: A = B*inv(B+D) where A is a square matrix, B a positive semi-definite matrix, D a positive diagonal matrix and inv() denotes the inverse matrix. All are real-valued. Does anyone know of any simple way to check whether this...

Homework Statement Find the exact values of A, B, and C in the following partial fraction decomposition.Then obtain the integral using those values. 1/(x^3-3x^2) = A/x + B/x^2 + C/(x-3)Homework Equations I'm not sure at this point.The Attempt at a Solution To make the denominator on the right...

I asked a question about Schmidt decomposition in one of the math forums, but haven't gotten any replies yet. Link. The question is about why the state of a bipartite system (a system that consist of two component systems) can be expressed in the form \sum_{n=1}^\infty...

I would be interested in seeing a correct statement and proof of the Schmidt decomposition theorem that (I think) says that if x\otimes y\in H_1\otimes H_2, there exists a choice of bases for these Hilbert spaces, such that x\otimes y=\sum_{n=1}^\infty \sqrt{p_n}\ a_n\otimes b_n with...

I have recently found that the half-life of ozone at a temperature of 250°C is 1.5 seconds. However, this source (http://www.lenntech.com/library/ozone/decomposition/ozone-decomposition.htm) failed to give any statistics on the effect of pressure on ozone decomposition. My question is; if the...

Homework Statement Sorry I don't have equation editor working 1/(z+1)(z2 + 2z + 2) Homework Equations The Attempt at a Solution (z2 + 2z + 2) z2 + 1) can be factor as (z - i)(z + i) However, I'm having trouble seeing the pattern on what (z2 + 2z + 2) would become, I...

If a n x n matrix A has an eigenvalue decomposition, so if it has n different eigenvalues, by the way, is it correct that a n x n matrix that doesn't have n different eigenvalues can't be decomposed? Are the more situations in which it can't be decomposed? Why can't I just put the same...

Hi, I am reading through Weinberg's "Quantum Theory of Fields" (vol. 1) and I am somewhat confused about the signs in the cluster decomposition of the S-matrix. Specifically, referring to eq. 4.3.2, let's say the term coming from the partition \alpha \to \alpha_1\alpha_2,\beta \to...

Homework Statement Prove that if A is an nxn positive definite symmetric matrix, then an orthogonal diagonalization A = PDP' is a singular value decomposition. (where P' = transpose(P))2. The attempt at a solution. I really don't know how to start this problem off. I know that the singular...

I was just wondering if it was possible to find the singular value decomposition of an nxn matrix such as 1 1 -1 1 I tried this but then when finding the eigenvectors of A^T*A I found there were none (non-trivial anyhow). So, is this not possible? EDIT: How embarrassing I made an...

The first-order decomposition of a colored chemical species, X, into colorless products is monitored with a spectrophotometer by measuring changes in absorbance over time. Species X has a molar absorptivity constant of 5.00 x 10^3 cm^ -1 M^ -1 and the path length of the cuvette containing the...

Hello, If I've a real signal, and I do a forward Fourier transformation I receive two parts: Real and Imaginary, what's the difference between them? i need to represent the transform in a software program, which part do i represent ?

How would I calculate the determinant of a square matrix using LU Decomposition. Please be plain, I am not good with technical terms. An example would be nice. Thank you!

Hi, if let's say that there's an even function f(z) then how do we know if its Laurent decomposition (i.e. f(z) = f0(z) + f1(z) ) will be even functions and have even powers of z? Any help will be greatly appreciated.

Homework Statement 1/[s*(s^2 + 4)] Homework Equations The Attempt at a Solution 1/[s*(s^2 + 4)] = A/(s) + (Bs + C)/(s^2 + 4) => 1 = A(s^2 + 4) + (Bs + C)s s = 0 1 = A(0 + 4) + (B*0 + C)*0 A = 1/4 s = i 1 = A(i^2 + 4) + (Bi + C)i 1 = A(i^2 + 4) + Bi^2 + Ci 1 =...

Homework Statement I just can't understand it I've read plenty of guides online, I just can't figure it out. How do you do partial fraction decomposition the farthest i can get is below 1/[(s^2 + 1)(s+1)] Homework Equations The Attempt at a Solution 1/[(s^2 + 1)(s+1)] = A/(s+1)...

Homework Statement An nxn matrix C is skew symmetric if C^t = -C. Prove that every square matrix A can be written uniquely as A = B + C where B is symmetric and C is skew symmetric. Homework Equations The Attempt at a Solution No clue.

Is entangle enough to find that given states have Schmidt decomposition?

Hello, I would greatly appreciate any comment on the following problem: Suppose that I estimate the spectral density of a weakly stationary series, say, nonparametrically by smoothing the periodogram. Is there a simple way to recover the coefficients of the wold decomposition of the process...

Homework Statement Find the decomposition of the standard two-dimensional rotation representation of the cyclic group Cn by rotations into irreducible representations The Attempt at a Solution Ok i did this directly, finding complementary 1-dimensional G-invariant subspaces. but...

I am just coming back to math after a, oh 30 year or so, vacation. In the class I'm taking, we are studying Partial Fraction Decomposition ( Px/Qx:Qx). It doesn't entirely make sense to me, tho like a monkey typing the great American novel, I can solve them given enough time. I am just having...

I'm trying to find a divergenceless vector field based on its curl, and discovered that I could use a http://en.wikipedia.org/wiki/Helmholtz_decomposition" , and the article I found on this didn't make much sense to me. First, can someone confirm that the dimension referred to in the...

I'm trying to find a divergenceless vector field based on its curl, and discovered that I could use a http://en.wikipedia.org/wiki/Helmholtz_decomposition" , and the article I found on this didn't make much sense to me. First, can someone confirm that the dimension referred to in the...

Wikipedia defines the Shur decomposition of matrix A as A = Q U Q^{-1} where Q is unitary and U is upper triangular. http://en.wikipedia.org/wiki/Schur_decomposition Mathworld defines the Shur decomposition of matrix A as Q^H A Q = T, where Q is unitary and T is upper...

Has there been any real scientific testing to determine the ratio of aging or decomposition of space junk above our atmosphere compared to atmospheric aging? I ask this because if space junk is going to be "out there" for millions of years as is, why would we even consider adding more in the...

Homework Statement Supposedly the process to solve Ax=0 is to solve Transpose(R).Ry=z (where z is a random vector) and then x=y/(norm-2 of y). Homework Equations The Attempt at a Solution Ay=b for some random vector b Transpose(A).Ay= Transpose(A).b...

Homework Statement Let A be a real or complex nxn matrix with Jordan decomposition A = X \Lambda X^{-1} where \Lambda is a diagonal matrix with diagonal elements \lambda_1,..., \lambda_n. Show that for any polynomial p(x): p(A)=Xp(\Lambda)X^{-1} p(\Lambda) should really be the matrix with...

Hello, I studying General Relativity at University of Chicago and I am looking for information regarding "Fourier-Harmonic Mode Decomposition" or mode decomposition in general and "Retarded Greens Functions" and retarded fields in general. Does anyone have advice about where is a good...

Homework Statement Let A be a complex or real square matrix. Suppose we have a Jordan decomposition A = XJX-1, where X is non-singular and J is upper bidiagonal. Show how you can obtain a Schur Decomposition from a Jordan Decomposition. Homework Equations Schur Decomposition: A = QTQ*...

Homework Statement Find the partial fraction decomposition of : \frac{x^2}{(1-x^4)^2} The Attempt at a Solution \frac{x^2}{(1-x^4)^2}=\frac{A}{(1-x^4)}+\frac {B}{(1-x^4)^2} =A(1-x^4)+B when x=1 1=A(1-1^4)+B Hence B=1 and A=0 \frac{x^2}{(1-x^4)^2}=\frac{0}{(1-x^4)}+ \frac{1}{(1-x^4)^2}...

Hi, I have a practical assessment tomorrow which deals with the Thermal Decomposition of a metal Carbonate. Is it possible to write Ionic Equations for the reaction? Ionic solids and solutions are involved, but the reactions seem to be Double Decomposition, especially the test for CO2 Also...

I'm supposed to prove this step as part of my proof for existence of Cholesky Decomposition. I can see how to use it in my proof, but I can't seem to be able to prove this lemma: For any positive (nxn) matrix A and any non-singular (nxn) matrix X, prove that B=X^{\dagger}A X is...

Homework Statement Let A be a real mxn matrix, m>=n, with singular values \sigmaj.Show that the singular values of (\stackrel{I_{n}}{A}) are equal to \sqrt{1+\sigma_j^2}. Homework Equations The Attempt at a Solution I know that an SVD for A is A = U(\stackrel{\Sigma}{0})v^T and...

First Order Decomposition -> % of compound. Please help! Homework Statement In a first order decomposition, the constant is 0.00313 sec-1. What percentage of the compound has decomposed after 7.17 minutes? Homework Equations rate = ln[A]o - ln[A]t = kt The Attempt at a Solution I...

Checkerboard decomposition? I need help from guys working with Quantum Monte Carlo. I'm reading about worldline method applied to Heisenberg chain (or nearest-neighbor Hubbard model) and after calculation of nonzero matrix elements I'm immediately thrown to the checkerboard representation of...

HELP! on Partial Fraction Decomposition Problem! - Heavy Side Technique I am doing a partial fraction decomposition problem for my calc 2 class. We use the Heavy Side Technique, but I will take help either way! \frac{2x-1}{x(x^2+1)^2} Thank you!

Say X is a CW-complex. Then for any n, the n-skeleton X^n of X is obtained from the (n-1)-skeleton X^(n-1) by gluing some n-cells on X^(n-1) along their boundary. From what I read, it seems that the way to obtain X^n from X^(n-1) in this way is not unique. Is this non-uniqueness superfluous...

Given random variables X and Y, which are not independent, is it always possible to find a random variable W which is independent from X, such that Y = f(X,W), for some function f? Example: let the joint distribution of X and Y be Y 0 1 X+------- 0|1/3 1/6 1|1/6 1/3 Then if we...

Homework Statement I need to find a Jordan decomposition for: \[ \left( \begin{array}{cccc} 2 & 0 & 1 & 2 \\ -1 & 3 & 0 & -1 \\ 2 & -2 & 4 & 6 \\ -1 & 1 & -1 & -1 \end{array} \right)\] Homework Equations The Attempt at a Solution I found the eigenvalues: 2 (m=4). I also found the eigenvectors...

Homework Statement Let A\inRmxn be factorized as A=QR where Q\inRmxn has orthonormal columns. Prove that A+=R+QT Homework Equations R is an upper triangular matrix The Attempt at a Solution I tried to apply the definition A+=(ATA)+A+ I ended up here: R+(RR+)TQT I'm not sure if...

[b]1. Let T be the linear operator on R^n that has the given matrix A relative to the standard basis. Find the spectral decomposition of T. A= 7, 3, 3, 2 0, 1, 2,-4 -8,-4,-5,0 2, 1, 2, 3 [b]3. eigen values are 1 (mulitplicity 1), -1 (mult. 1), 3 (mult. 2). And associated eigen...

I am looking for a mechanism to find a decomposition of symmetric groups. For finitely generated abelian group G, there is a mechanism to decompose G such that G is isomorphic to a direct sum of cyclic groups. For symmetric groups, it seems a bit complex for me to find it. For example...

Homework Statement Suppose V is a complex vector space and T \in L(V). Prove that there does not exist a direct sum decomposition of V into two proper subspaces invariant under T if and only if the minimal polynomial of T is of the form (z - \lambda)^{dim V} for some \lambda \in C. Homework...

Homework Statement let a=(b_1,...,b_n) n-cycle in the permutation group S_n . prove that the cycle decomposition of a^k consist of gcd(n,k) cycles of n/gcd(n,k) size. The Attempt at a Solution I know that a^k(b_i)=b_{i+k (mod n)} how can it help me ?

Homework Statement \frac{4x^{4}-8x^{3}+5x^{2}-2x-1}{2x^{2}-3x-2} Homework Equations The Attempt at a Solution I started of by breaking the bottom part down into (2x+1)(x-2) which then allowed me to set... \frac{A}{(2x+1)}+\frac{B}{(x-2)} The problem is from here I tried...

Homework Statement http://img296.imageshack.us/img296/6584/93872041bz2.png Homework Equations http://img208.imageshack.us/img208/2062/55585340nj2.png The Attempt at a Solution I think the key to the solution is the matrix A being non-singular but I can't see how.

In finite dimensions, a matrix can be decomposed into the sum of rank-1 matrices. This got me thinking - in what situations can a bounded linear operator mapping between infinite dimensional spaces be written as an (infinite) sum of rank-1 operators? eg, let A be a bounded linear operator...

What I'm wondering is: Q and R in the QR decomposition of A are the same Q and R in the RQ decomposition of which matrix? I found some MATLAB code which will get RQ from QR, but I don't understand how you would do those operations FIRST, then find the QR decomposition. ReverseRows = [0...