Symmetric Definition and 539 Threads

This property is given in my book. The square of any determinant is a symmetric determinant. Well it works when I take a determinant say 3x3 and multiply it by itself using row to row multiplication. But it fails if I multiply using row to column. Thanks

I'm looking for the general form of a symmetric 3×3 matrix (or tensor) ##\textbf{A}## with only two different eigenvalues, i.e. of a matrix with the diagonalized form ##\textbf{D}=\begin{pmatrix}a& 0 & 0\\0 & b & 0\\0 & 0 & b\end{pmatrix} = \text{diag}(a,b,b)##. In general, such a matrix can be...

I am more familiar with quantum physics than cosmology so it occurred to me that I hadn't heard anyone talk about the question of symmetry with respect to dark matter. If its influence is restricted to the disc of a galaxy, but it is particle-like in structure than symmetry is violated isn't...

Homework Statement $$\begin{bmatrix} a_{11} & a_{12} & 0 & 0\\ a_{12} & a_{22} & a_{23} & 0\\ 0 & a_{23} & a_{33} & a_{34} \\ 0 & 0 & a_{34} & a_{44} \\ \end{bmatrix} = \begin{bmatrix} q_{11} & q_{12} & q_{13} & q_{14} \\ q_{21} & q_{22} & q_{23} & q_{24} \\ q_{31} & q_{32} & q_{33} & q_{34}...

Homework Statement For the characteristic curves shown below, select resistors for an H-biased common emitter amplifier for symmetric operation. Show the load line and operating pointing on the graph Homework Equations What does it mean by "symmetric operation"? The Attempt at a Solution...

Homework Statement Suppose X is a set with n elements. Prove that Bij(X) ≅ S_n. Homework Equations S_n = Symmetric set ≅ = isomorphism Definition: Let G and G2 be groups. G and G2 are called Isomorphic if there exists a bijection ϑ:G->G2 such that for all x,y∈G, ϑ(xy) = ϑ(x)ϑ(y) where the...

Disclaimer: This is a homework problem I need to analytically solve the diffusion equation for a 1d 1 group slab with width a, and source distribution Se^(-k(x+a/2)) I've gone through the math, and come up with my homogeneous and particular solution and attempted to apply the boundary...

Hi, I've been trying to find one symmetric "injective" N²->N function, but could not find any. The quotes are there because the function I'm trying to find is not really injective, as I need that the two arguments be interchangeable and the value remains the same. In other words, the tuple (a...

Express r12+r22+...+rn2 as a polynomial in the elementary symmetric polynomials s1, s2, . . . ,sn. I'm sure the equation we are dealing with is (r1+r2+...+rn)2 which is very large to factor out but should yield r12+r22+...+rn2+(other terms) I believe s1=r1+r2+...+rn s2=Σri1ri2 for...

Question: Let τ = (i, j) ∈ Sn with 1 ≤ i < j ≤ n. Prove: δ(rτ(1) , . . . ,rτ(n) ) = −δ(r1, . . . ,rn) Note: Discriminant Polynomial δ(r1,r2,...,rn) = ∏ (ri - rj) for i<j I am pretty confused on where to begin. Based on the note, does −δ(r1, . . . ,rn) then =...

Problem: In $S_3$ give an example of two elements $x$ and $y$ such that $(xy)^2 \ne x^2y^2$. Attempt: Consider the mapping $\phi: x_1 \mapsto x_2, x_2 \mapsto x_1, x_3 \mapsto x_3$ and the mapping $\psi: x_1 \mapsto x_2, x_2 \mapsto x_3, x_3 \mapsto x_1$. We have that the elements $\phi, \psi...

Is it possible for a nucleus to undergo antisymmetric spontaneous fission? And if so, what is the process responsible? Thanks

Homework Statement Homework Equations /The Attempt at a Solution[/B] I am trying to solve problem 2-13 from my textbook "Principles of Electrodynamics" (see image below). I believe that I should be solving the potential as \varphi(r,\theta) = \sum_{n=0}^\infty (A_n r^n +...

In case of the Green's functions for the Laplace equation, we know that they're all symmetric under the exchange of primed and un-primed variables. But is it generally true for the Green's functions of all differential equations? Thanks

The following extract is taken from Appendix A of the following paper: http://arxiv.org/abs/0810.0713.Any solution of the Schrodinger equation with rotational invariance around the ##z##-axis can be expanded as ##\psi_{k}=\Sigma_{l}A_{l}P_{l}(cos \theta)R_{kl}(r)##, where ##R_{kl}(r)## are the...

Homework Statement I think, to normalize a wavefunction, we integrate over the solid angle ##r^2 dr d\theta d\phi##. Typically we have ## R(r)Y(\theta, \phi) ## as solutions. If ##Y## is properly normalized, then the normalization condition for ##R(r)## ought to be $$ \int_0^\infty dr r^2...

Homework Statement I'm reading something in my quantum physics book that says given a wavefunction ψ that is even, if we evaluate its integral from -ε to ε, the integral is 0. How can this be? I thought this is the property of odd functions. Homework Equations ψ=Aekx if x<0 and ψ=Be-kx if x>0...

I think it's 3... All 2x2 can be written as a_1 A_1 + a_2 A_2 + a_3 A_3 + a_4 A_4 with A_1 = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} , A_2 = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} , A_3 = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} , A_4 = \begin{bmatrix} 0 & 0 \\ 0 & 1...

In a spherical symmetric problem the only nonzero components of the electric and the magnetic field are Er and Br Why?

Homework Statement Let K and L be symmetric PSD matrices of size N*N, with all entries in [0,1]. Let i be any number in 1...N and K’, L’ be two new symmetric PSD matrices, each with only row i and column i different from K and L. I would like to obtain an upper bound of the equation below...

Hi, I was wondering how to find a minimal set of generators for the symmetric groups. Would it be difficult to fill-in the following table? ##\begin{array}{cl} S_3&=\big<(1\;2),(2\;3)\big> \\ S_4&=\big<(1\;2\;3\;4),(1\;2\;4\;3)\big>\\ \vdots\\ S_{500} \end{array} ## Is there a procedure to...

First by "this derivation" I'm referring to an online tutorial: http://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node9.html It's said in the above tutorial that the ##i-th## component of the total torque acting on a fluid element is ##\tau_i = \int_V \epsilon_{ijk} \cdot x_{j} \cdot F_{k}...

I was looking at this excellent website this afternoon, and was puzzled by two diagrams showing the symmetric and antisymmetric wavefunctions. In the latter case the text states that the particles are far away from each other, explaining the Pauli Exclusion Principle, etc. But looking at the...

It seems there should be a list of tensor identities on the internet that answers the following, but I can't find one. For tensors in ##R^4##, ##S = S_\mu{}^\nu = S_{(\mu}{}^{\nu)}## is a symmetric tensor. ##A = A_{\nu\rho\sigma}= A_{[\nu\rho\sigma]}## is an antisymmetric tensor in all...

A friend of mine heard a popular science show on the radio. A caller asked what is better to wear on a hot day, white clothes or black clothes. The answer given was that it did not matter because although black absorbs more readily it also radiates it more readily. My friend said of course that...

Homework Statement A particle of mass m moves in a "central potential" , V(r), where r denotes the radial displacement of the particle from a fixed origin. From Hamilton´s equations, obtain a "one-dimensional" equation for {\dot p_r}, in the form {{\dot p}_r} = - \frac{\partial }{{\partial...

Hello I was reading Spivak's calculus. It starts with discussing the familiar axioms of the real numbers. He calls them properties. At some another forum, I came across the reference to Landau's "Foundation of Analysis" as a background for analysis. So I referred to that book. On the very first...

Homework Statement Consider the following inital states of the symmetric 2D harmonic oscillator ket (phi 1) = 1/sqrt(2) (ket(0)_x ket(1)_y + ket (1)_x ket (0)_y) ket (phi 2) = 1/sqrt(2) (ket(0)_x ket(0)_y + ket (1)_x ket (0)_y) Calculate the <p_x (t)> for each state Homework EquationsThe...

Hello,Suppose I have a vector equation: \begin{cases} x=0+10t\\ y=0+10t\\ z=0+10t \end{cases} Which forms the symmetric equation \frac{x-0}{10}=\frac{y-0}{10}=\frac{z-0}{10} Now, I know the symmetric equations can be split up so that you can form the two planes whose intersection yields the...

I am having problem working with the objects in the title. Working with permutations, rotations and reflections is fine, but I have problem with the following: Showing a subgroup is or is not normal (usually worse in the case of symmetric groups) Finding a subgroup of order n. Showing that...

As a step in a solution to another question our lecture notes claim that the matrix (a,b,c,d are real scalars). \begin{bmatrix} 2a & b(1+d) \\ b(1+d)& 2dc \\ \end{bmatrix} Is positive definite if the determinant is positive. Why? Since the matrix is symmetric it's positive definite if the it...

In most GR textbooks, the general form of a spherically symmetric metric is obtained by inspection which is acceptable. But in the textbook I'm reading, the author does that with a mathematical analysis just to illustrate the method. But I can't follow his calculations. In fact he omits much of...

I need to find the vectors for time and radius that describe a space-like 4-acceleration of an observer falling radially into a spherically-symmetric black hole. Previous to this question, the values of the real time derivatives for time and radius were derived to be: dt/dτ = (1-2m/r)-1 and...

I'm hoping someone can clarify for me, I have seen the following used: \frac{\partial}{\partial g^{ab}}\left( g^{cd} \right) = \frac{1}{2} \left( \delta_a^c \delta_b^d + \delta_b^c \delta_a^d\right) I understand the two half terms are used to account for the symmetry of the metric tensor...

Homework Statement The question is how to get the mass term for the W Gauge bosons within the Left Right Symmetric Model (LRSM). My main struggle is with the Left-Right Mixing part. So the LRSM has the gauge group $SU(2)_L\times SU(2)_R \times U(1)_{B-L}$ and with an additional discrete...

Hi, I'k looking at some MATLAB code specifically eig2image.m at: http://www.mathworks.com/matlabcentral/fileexchange/24409-hessian-based-frangi-vesselness-filter/content/FrangiFilter2D So, I understand how the computations are done with respect to the eigenvector / eigenvalues and using...

Homework Statement Consider a symmetric n x n matrix ##A## with ##A^2=A##. Is the linear transformation ##T(\vec{x})=A\vec{x}## necessarily the orthogonal projection onto a subspace of ##R^n##? Homework Equations Symmetric matrix means ##A=A^T## An orthogonal projection matrix is given by...

Hi, I want to show that the Trace of the Product of a symetric Matrix (say A) and an antisymetric (B) Matrix is zero. $So\: (AB)_{ij}=\sum_{k}^{}{a}_{ik}{b}_{kj} $ $and\: Tr(AB)=\sum_{i=j}^{}(AB)_{ij}=\sum_{i}^{}\sum_{k}^{}{a}_{ik}{b}_{ki} $ $because\:A\:is\:symetric, \: {a}_{ik}=...

Would this example be valid in satisfying a relation that is symmetric and anti-symmetric? The relation R = {(1,1),(2,2)} on the set A = {1,2,3} Also, I'm curious to know since relations can both be neither symmetric and anti-symmetric, would R = {(1,2),(2,1),(2,3)} be an example of such a...

I stumbled upon this article: http://www.comsol.com/blogs/exploiting-symmetry-simplify-magnetic-field-modeling/ Since the article does not contain any mathematical formulations, I was wondering how the boundary conditions can be expressed in terms of magnetic vector potential. From what I...

Homework Statement Homework Equations Relabelling of indeces, 4-vector notation The Attempt at a Solution The forth line where I've circled one of the components in red, I am unsure why you can simply let ν=μ and μ=v for the second part of the line only then relate it to the first part and...

If σ is a k-cycle with k odd, prove that there is a cycle τ such that τ^2=σ. I know that every cycle in Sn is the product of disjoint cycles as well as the product of transpositions; however, I'm not sure if using these facts would help me with this proof. Could anyone point me in the right...

So I was bored in math class and came up with this series of related questions, that I cannot answer: Is there a clean expression for $f'(x),$ where $$f(x)=\prod_{i=1}^{n}\dfrac{(x-i)}{(x+i)}?$$ What about for $f''(x)?$ Or for $$f(x)=\prod_{i=1}^{n}\dfrac{(x^2-i)}{(x^2+i)}?$$

I'm looking at Lecture Notes on General Relativity, Sean M. Carroll, deriving the FRW metric, and I'm a little confused with the use of eq 8.4 . I thought instead it should be using the general form of a spherically symmetric metric, not a vacuum spherical symmetric - eq 7.13 has been derived...

Hi, Well I hope it's not a thread that already is in the storage here. I want to understand the image of a matrix. not only calculating it but also why I'm doing that. Here are my questions: 1) They say d = Lx has a solution if d ∈ ImL. I know that the image of a matrix is calculated by Lx = x...

Why is the function: y = f(x) = a(x-1)(x-2)(x-3) symmetrical about the line x = 2? I mean how can we be sure that it is? Is there any method to check it?

In the text I'm looking at, the Schwarzschild metric derivation, and it argues to the form ## ds^{2}= -e^{2\alpha(r)} dt^{2} + e^{\beta(r)}+r^{2}d\Omega^{2} ## [1]. Up to this point some of the ##R_{uv}=0## components have been used, not all. It then says we have proven any spherically symmetric...

This is probably a stupid question, but, is the Schwarzschild metric spherically symmetric just with respect to space or space-time? Looking at the derivation, my thoughts are that it is just wrt space because the derivation is use of 3 space-like Killing vectors , these describe 2-spheres, and...

I guess that nature do not care about "directions". If I'm performing an experiment on (for example) light the result is invariant on the direction of the laboratory. So why we observe directionality on the nature that we are observing around us? Why "things" are definitively not symmetric...

Homework Statement (a) Find the spatial wavefunction (b)Show anti-symmetric wavefunctions have larger mean spacing (c) Discuss the importance of this (d)Determine the total orbital angular momentum (e)Hence find the ground state term for Z=15[/B] Homework EquationsThe Attempt at a Solution...