Decomposition Definition and 386 Threads

If you have a collection of n (nonzero and different) eigenvectors, is there a way to find a general form of an n-by-n matrix that corresponds to them in such a way that 'rules out' alternative forms? For example, let's say we have the vectors ##\begin{bmatrix}c\\1\end{bmatrix}## and...

Let's construct a model: a small ball with mass m is thrown on a horizontal plane with a V, and the direction of the velocity makes an angle θ with the horizontal plane. What is the maximum height it can reach? (ignoring air resistance) We that the kinetic energy of an object will not be...

Using York decomposition as a foundation, can we reformulate General Relativity to treat space as flat (##R_{ij} = 0##), redistributing all curvature into a vector field for time (##\vec{t}_i##)? The York decomposition: $$\tilde{h}_{ij} = \phi^{-4} h_{ij}$$ $$\tilde{K}_{ij} = \phi^2 K_{ij}$$...

If bones were laid bare in a room at typical room temperatures for say southern Italy, how long would human bones take to decay naturally? So, not touched, in open air inside a room, and typical weather conditions. Let's assume the doors are closed, but there is no insulation.

So i'm doing a project for school with the local firefighters and we have to help save time to rescue people from a fire. So i had an idea, What if we made a transportable device that could contain the chemicals and stuff to create a decomposition reaction. I wanted to test my theory that...

$$\lambda_x = \frac{L}{n_x} , \lambda_y=\frac{L}{n_y} . \lambda$$ does not fit the cube as integer. In the figure $$n_x=4, n_y=3$$

I was reading Dunne's review paper on Chern-Simons theory (Les-Houches School 1998) and I don't get how he decomposes the gauge potential on the torus. My own knowledge of differential geometry is sketchy. I do know that the Hodge decomposition theorem states that a differential form can be...

Using the QR decomposition (the complex version) I want to prove that ##SL_2(C)## is homeomorphic to the product ##SU(2) × T## where ##T## is the set of upper-triangular 2×2-complex matrices with real positive entries at the diagonal. Deduce that ##SL(2, C)## is simply-connect. So, I can define...

For this problem, Find ##A^{-1}## given, The solution is, However, in the first image, why are we allowed to put together the submatrices in random order? In general does someone please know why we are allowed to decompose matrices like this? Many thanks!

New to group theory. I have 3 questions: 1. Tensor decomposition into Tab = T[ab] +T(traceless){ab} + Tr(T{ab}) leads to invariant subspaces. Is this enough to imply these subreps are irreducible? 2. The Symn representations of a group are irreps. Why? 3. What is the connection between...

## A = \pmatrix{ -4 & -3 & 3 & 3 \\ -3 & -4 & 3 & 3 \\ -6 & -3 & 5 & 3 \\ -3 & -6 & 3 & 5 } ## over ## \mathbb{R}##. Let ## T_A: \mathbb{R}^4 \to \mathbb{R}^4 ## be defined as ## T_A v = Av ##. Thus, ## T_A ## represents ## A ## in the standard basis, meaning ## [ T_A]_{e} = A ##. I've...

Suppose an optical scalar wave traveling in Z direction. Using the diffraction theory of Fourier Optics, we can predict its new distribution after a distance Z. The core idea of Fourier Optics is to decompose a scalar wave into plane waves traveling in different directions. But this...

I'm looking to check my understanding of the information below and ultimately get a better understanding of it. Is spectral decomposition a mathematical procedure? Does "the state space of the measured system" refer to the possible values that the system could take, when measured?

My Answer: I am still beginner in this area so it s quite hard for me to understand this one. I am not sure what the output that this question asked me. I thought it might be asked about the value of x1, x2, x3, and x4

Say you want to find the following Integrals $$\int \frac{1}{(x-1)(x+2)} (dx)$$ $$\int \frac{1}{(x-1)(x^2 + 2)} (dx)$$ The easiest way to solve them will be by using partial fraction decomposition on both the given functions. Decomposing the first function, $$\frac{1}{(x-1)(x+2)} =...

Hello, I am currently studying the Schmidt decomposition and how to use it to determine if a state is entangled or not and I can't understand how to write the state as a matrix so I can apply the Singular Value Decomposition and find the Schmidt coefficients. The exercise I am trying to complete...

Hello! Im having some trouble with solving ODE's using Laplace transformation,specifically ODE's that require partial fraction decomposition.Now I know how to do partial fraction decomposition,and have done it many times on standard polynoms but here some things just are not clear to me.For...

Hey everybody! I have put G1 = (1-s)/(2-10s) & G2 = (2-10s)/ (2 +10s) but than I read that all poles and zeroes should be inside the unit circle, and I don't know how to move the Zero S_01 = 1 to the unit circle

In a recent thread, I said that if there was interest, I would post in a separate thread the calculations for the kinematic decomposition of the congruence of worldlines describing the rod in the "rod and hole" relativity paradox discussed in that thread. Since there was interest, I am posting...

So the acceleration of point A was given by a force F exerted on cylinder that's along the direction of the stick, decomposed into the horizontal direction. so aA = F cos Θ The same force along the opposite direction is exerted on stick, and if we decompose that in vertical and horizontal...

It's a puzzle. I have decomposed vector v by using formulas known from physics: m*g*sin(theta) and m*g*cos(theta). I got: ##\vec v = (5, 5*\sqrt{3})## But it has been marked as wrong. Consequently, the rest of my calculations is not correct. Could you tell me, why?

Hey! :giggle: At the QR-decomposition with permutation matrix is the matrix $R$ equal to $R=G_3^{-1}P_1G_2^{-1}P_0G_1^{-1}A$ or $G_3P_1G_2P_0G_1A=R$? Which is the correct one? Or are these two equivalent? In general, it holds that $QR=PA$, right? :unsure:

My solution: partial pressure of C5H6O3 = mRT/MV = (5.63 g)(0.08206 L*atm/mol*K)(473 K) / (114.098 g/mol)(2.50 L) = 0.766 atm equilibrium partial pressure of C5H6O3 = 0.766 - x equilibrium partial pressure of C2H6 = x equilibrium partial pressure of CO = 3x total pressure = 0.766 atm - x + x +...

Let's say I want to study subalgebras of the indefinite orthogonal algebra ##\mathfrak{o}(m,n)## (corresponding to the group ##O(m,n)##, with ##m## and ##n## being some positive integers), and am told that it can be decomposed into the direct sum $$\mathfrak{o}(m,n) = \mathfrak{o}(m-x,n-x)...

I think this is a first order reaction because ln[C12H22O11] vs. time is linear. The k value is the negative of the slope. Therefore, my answer is rate = 0.45 hr-1 [C12H22O11]. The correct solution is rate = -0.45 hr-1 [C12H22O11]0. I don't understand why this is a zero order reaction, or why...

The solution says that when the effusion rate ratio is multiplied by the equilibrium mole ratio of H2 to CH3OH, the effused mixture will have 33.0 times as much H2 as CH3OH. I don't understand why. I just set the equilibrium mole ratio of H2 to CH3OH as equal to 33.0 times, Why is this...

Ca(HCO3)2 -> CaCO3 + H2O + CO2 First I evaluate the moles of calcium carbonate (don't mind the units, just to save time) ##\frac {80.0}{40,00+12.01+3*16,00}= 0,799 mol## From the equation, correct me if I am wrong , one mole of CaCO3 is proportional to one mole of CO2, so from this I can...

Hi, suppose I am given an SL(2C) matrix of the form ##\exp(i\alpha/2 \vec{t}\cdot\vec{\sigma})## where ##\alpha## is the complex rotation angle, ##\vec{t}## the complex rotation axis and ##\vec{\sigma}## the vector of the three Pauli matrices. I would like to decompose this vector into...

The following three equations illustrate this decomposition. Estimate separate linear wage regressions for individuals i in groups A and B: {\displaystyle {\begin{aligned}(1)\qquad \ln({\text{wages}}_{A_{i}})&=X_{A_{i}}\beta _{A}+\mu _{A_{i}}\\(2)\qquad...

Hey! 😊 I saw the below sentence in some notes: Let $A\in \mathbb{R}^{n\times n}$ be a not necessarily symmetric, strictly positive definite matrix, $x^TAx>0$, $x\neq 0$ und $Q\in \mathbb{R}^{n\times n}$ an orthogonal matrix, then $B=Q^TAQ$ has a LU decomposition. I want to understand...

Hey! 😊 We consider the matrix $$A=\begin{pmatrix}1 & -2 & 5 & 0 & 5\\ 1 & 0 & 2 & 0 & 3\\ 1 & 2 & 5 & 4 & 6 \\ -2 & 2 & -4 & 1 & -6 \\ 3 & 4 & 9 & 5 & 11\end{pmatrix}$$ I want to find the LU decomposition. How do we know if we have to do the decomposition with pivoting or without? :unsure:

According to the book I am using, one can decompose a finite abelian group uniquely as a direct sum of cyclic groups with prime power orders. Uniquely meaning that the structures in the group somehow force you to one particular decomposition for any given group. Unfortunately, the book gives no...

The eigen wavelengths λn(WL) of EM radiation in box are 2d/n where d is the size of the box. If I put a photon in a box with WL>2d via an optic cable trough a hole it must reflect on the perfect mirror walls and be a running wave. Maybe it is possible to decompose it as a set of eigenmodes of...

Hey! 😊 Let $A$ a $n\times n$ matrix with known LU decomposition, let $u\in \mathbb{R}^n, v\in \mathbb{R}^{n+1}$. Show that the number of multiplications and divisions that are needed to get a LU decomposition of the $(n+1)\times (n+1)$ matrix $$\begin{pmatrix}A & u \\ v^T\end{pmatrix}$$ is at...

Hey! 😊 Let $A=L^TDL$ be the Cholesky decomposition of a symmetric matrix, at which the left upper triangular $L$ hat only $1$ on the diagonal and $D$ is a diagonal matrix with positiv elements on the diagonal. I want to show that such a decomposition exists if and only if $A$ is positive...

Hey! :o We have the following permutations in $\text{Sym}(14)$ : - $\pi_1=(1 \ 2\ 4 \ 9)\circ(1 \ 3)\circ (6 \ 8\ 12)$ - $\pi_2=(2 \ 4\ 5 \ 8\ 7)\circ (1 \ 12 \ 6)\circ \ (13 \ 14)$ - $\pi_3=(1 \ 4 \ 5\ 8 \ 11)\circ (2 \ 4\ 6 \ 5 \ 1)$ 1. Determine the cycle decomposition of $\pi_1...

Hi at all! I need to implement the Pivoted Cholesky Decomposition in C++ and I know that is possible implement it without rows permutations. Where can I find the algorithm described clearly and/or codes example in other language to replicate in C++? Thanks!

Hi, I understand how any function could be decomposed into even and odd parts assuming the function isn't a purely even or odd to start with. It's just like saying that any vector in x-y plane could be decomposed into its x- and y-component assuming it doesn't lie parallel to x- or y-axis...

Hi at all, I have to calculate the Cholesky decomposition of a symmetric matrix and this is the C ++ code I wrote: boost::numeric::ublas::matrix<double> Math::cholesky(const boost::numeric::ublas::matrix<double> &MatrixA) { int dim = MatrixA.size1()...

I am studying QFT from A First Book of QFT. It is a very well-written book. However, due to some personal reasons, I cannot buy the printed book at this moment. So I borrowed this book from a person (who, in turn, borrowed it from his university library), and scanned it. Everything is fine...

Hello! There is a problem to write chemical reactions that goes with substances if they are not stored properly. For example theophylline should be saved from light and though I am trying to find its’ reaction with hv(light) but failed. Please help with some good reference Many thanks in advance

If you see the $\sum \tau_0 = L\cdot N_1 \cdot cos \theta - LF_1 sin \theta - L/2 \cdot G cos \theta$, all the trigonemetric parts are all opposite of what i can understand, given the angle as drawed in the Picture/url. Please help me :)https://pasteboard.co/IiXr8qA.png

For my base case I just used a graph with three vertices and 2 edges. Decomposing this would just give us the same graph, which has a path length of 2. The inductive step is where I'm having some trouble: One idea I have is that we take a graph G then inductively remove an edge to create two...

Hi, there. I have some problems when learning Schmidt decomposition in Nielsen's QC. The statement of Schmidt decomposition is simple and clear, however, the book doesn't give a clear procedure to do the Schmidt decomposition. I don't know whether the proof under the theorem is the the one I...

Here is the initial matrix M: M = \begin{bmatrix} 3 & 1 & 6 \\ -6 & 0 & -16 \\ 0 & 8 & -17 \end{bmatrix} I have used the shortcut method outlined in this youtube video: LU Decomposition Shortcut Method. Here are the row reductions that I went through in order to get my U matrix: 1. R_3 -...

I think the cluster decomposition states that products of space like separated observable decouple when sandwiched with states. An analogy with statistical mechanics seems to suggest that we are stating there are no phase transitions. For example, in the Ising model all spins are correlated in...

From Weinberg, The Quantum Theory of Fields, Vol. 1, there is the statement that "the only way" to merge Lorentz invariance with the cluster decomposition property (a.k.a. locality) is through a field theory. He uses this argument basically to justify that any quantum theory at low energies...

It is common to write e.g photon two point function in terms of manifest transverse and longitudinal form factors with lorentz structure factored out, e.g $$\Pi^{\mu \nu} = (g^{\mu \nu} - q^{\mu} q^{\nu}/q^2)T_T + q^{\mu} q^{\nu}T_L,$$ where mu and nu are polarisation indices. How do I relate...

If the velocity gradient decomposition is done by symmetric and antisymmetric parts then ##\frac{\partial v^i}{\partial x^j}=\sigma_{ij}+\omega_{ij}## where ##\sigma _{ij}=\frac{1}{2}(\frac{\partial v^i}{\partial x^j}+\frac{\partial v^j}{\partial x^i})## and...

Can anyone illustrate for me matrix decomposition in a simple way?