Matrix Definition and 1000 Threads

Suppose I have some arbitrary square matrix M, and I want to build a unitary matrix U: U=\left[\begin{array}{c|c}M & N \\\hline O & P\end{array}\right] Does there exist some general procedure for determining N, O, and P given M?

I have known: (1) A Hamiltonian, say, H(k), where k is the crystal momentum. (2) An appropriate complete basis set {a_1,a_2,a_3…}. (3) Some symmetric operators {A,B,…} which commute with H(k), i.e. [A,H]=[B,H]=...=0. Of course, by calculation, I can get any matrix element of H(k), i.e...

Homework Statement Using the following information, find the matrix A (I+2A)-1 = [-1 2] [4 5] Homework Equations AA-1 = I The Attempt at a Solution none. I have no idea how should I start. The inverse on the whole left side is driving me crazy.

Hi there. The question I wanted to ask is: Why are matrix methods so widely used for numerical solution of partial differential equations? Many times I've found that storing a whole matrix requires much more memory than just doing an iteration scheme to propagate the solution. Sometimes I...

Why is the dot product equivalent to the matrix multiplication of its components? I've seen some proofs using Pythagorean and cosine law but they don't give you an intuitive feel as to why matrix multiplication works. The geometric definition (##ab cosθ##) is very easy to understand. To a...

Homework Statement Define a simple random walk Yn on a finite state space S = {0, 1, 2, . . . , N} to be a random process that • increases by 1, when possible, with probability p, • decreases by 1, when possible, with probability 1 − p, and • remains unchanged otherwise. (a) Specify the...

Homework Statement [/B] The population is 50 The diseases is known to follow SIS dynamics with the following probabilities The number of infected individuals increases with probability 0.1 and it decreases with probability 0.05 the probability that nothing happens is 0.85 a) what is the...

Homework Statement If ##A## is an ##n \times n## nilpotent matrix, then the characteristic polynomial of ##A## is ##x^n## Homework EquationsThe Attempt at a Solution Suppose that ##A## has an eigenvalue with corresponding eigenvector such that ##A v = \lambda v##. Then ##A^k v = \lambda^k v =...

If given a position vector defined for a orthogonal curvilinear coordinate system HOW would the matrices that make up the Levi Civita 3x3x3 matrix remain the same? "Levi Civita 3x3x3 is said to be independent of any coordinate system or metric...

Homework Statement Create two 5x5 arrays, A & B, and ask the person to fill them out. Save those numbers in matrix_a.txt & matrix_b.txt respectively. Then, save the sum and difference of those numbers in sum.txt & diff.txt respectively. Basically we need to create two arrays, fill them out...

Homework Statement Hi guys, I am having an issue understanding what to do with this question. The question is displayed below: I have hand wirtten my working, as I don't now how to do matrices fully on latext. I used the definition to get this far for part a, but not sure about the second...

So the question is show that $$S=\left\{ \begin{pmatrix} a & b\\ -b & a \end{pmatrix} :a,b \in \Bbb{R} ,\text{ not both zero}\right\}$$ is isomorphic to $\Bbb{C}^*$, which is a non-zero complex number considered as a group under multiplication So I've shown that it is a group homomorphism by...

Homework Statement Find all components of the matrix eiaB. a is a constant and B is a 3x3 matrix whose first row is 0,0,-i second row is 0,0,0 and third row is i,0,0. The taylor expansion of eiaB gives 1+iaB-a2B2/2! - ... Homework Equations The taylor expansion of eiaB gives 1+iaB-a2B2/2! -...

Homework Statement 3.For which values of ##\lambda## does the following system of equations also have non trivial solutions Homework EquationsThe Attempt at a Solution What I tried doing first is to put all variables on the same side and got ## v+y-\lambda*x=0\\ x+z-\lambda*y=0\\...

I have a set of k-points, e.g. k1,k2,k3,k4. and they are equivalent by symmetry. Now I have calculated the momentum matrix element <i|p|j> at k1 point ONLY, and then calculate the optical properties which, for example, depend on <i|p|j><j|p|i> I have to make a summation on four k-points...

I have this Hamiltonian --> (http://imgur.com/a/lpxCz) Where each G is a matrix. I want to find the eigenvalues but I'm getting hung up on the fact that there are 6 indices. Each G matrix lives in a different space so I can't just multiply the G matrices together. If I built this Hamiltonain...

I'm trying to solve a problem where I am given a few matrices and asked to determine if they could be density matrices or not and if they are if they represent pure or mixed ensembles. In the case of mixed ensembles, I should find a decomposition in terms of a sum of pure ensembles. The matrix...

Hi, got a question I'm stuck on.. Write down a matrix P which will diagonalise A and write down the corresponding diagonal matrix D, where D = P^-􀀀1AP. You do not have to calculate P^-1 Ive got all the eigenvalues and eigenvectors for A, and thus have the Matrix P, which has a determinant of...

This is a problem from Lang's Introduction to Linear Algebra. The problem statement is: Find a 2 x 2 matrix A such that A2= ##\begin{pmatrix} -1 & 0 \\ 0 & -1 \\ \end{pmatrix}## = -I The solution is available in the answer section of the book, but it is not shown how the author comes up with...

Homework Statement Let ##A## be an ##m \times n## matrix with rank ##m##. Prove that there exists an ##n \times m## matrix ##B## such that ##AB= I_m## Homework EquationsThe Attempt at a Solution So here is how far I get. I am given that ##A## has rank ##m##. Since ##L_A(x) = Ax## is a map...

Homework Statement Prove that if rank(A) = 0, then A = 0. Homework EquationsThe Attempt at a Solution This seems like a very easy problem, but I just want to make sure I am not missing anything. rank(A) = dim(Im(A)) = 0, so Im(A) = {0}. Thus, A is by definition the zero matrix. My only...

Homework Statement Find the Jordan canonical form of the matrix ## \left( \begin{array}{ccc} 1 & 1 \\ -1 & 3 \\ \end{array} \right)##. Homework EquationsThe Attempt at a Solution So my professor gave us the following procedure: 1. Find the eigenvalues for each matrix A. Your characteristic...

Hi, I have attached a pdf which shows clearly how I have carried out my transformations from one axis into another. However, I am not convinced that it is right and I have described why I feel so. I shall be grateful if someone can help me Kajal

Homework Statement Homework Equations ml2/12 The Attempt at a Solution So according to my databook: The axis x'-x' in the question corresponds to axis ZZ in the databook image above. That means in terms of radius, the moment of inertia about axis x'-x' is mr2/2. So in light of this, why...

Hello, What could be wrong when the total inertia matrix of a robotic manipulator is non invertible when under certain values of the joint angles? Thank you

How do we solve the particle in a box (infinite potential well) problem using matrix mechanics rather that using Schrodingers Equation? Schrodingers Equation for this particular problem is a simple partial differential equation and is easy for me to follow. The solution has the following...

Homework Statement how to find this Homework Equations none The Attempt at a Solution determinant??

Homework Statement Let ##B_1 = {\begin{bmatrix} 1 \\ 1 \\ 1\\ 0 \end{bmatrix}}, {\begin{bmatrix} 1 \\ 1 \\ 0\\ 0 \end{bmatrix}}, {\begin{bmatrix} 0 \\ 0 \\ 1\\ 1 \end{bmatrix}} ## and ##B_2 = {\begin{bmatrix} 1 \\ 1 \\ 1\\ 1 \end{bmatrix}}, {\begin{bmatrix} 1 \\ 1 \\ 1\\ -1 \end{bmatrix}}...

Homework Statement Write the density operator $$\rho=\frac{1}{3}|u><u|+\frac{2}{3}|v><v|+\frac{\sqrt{2}}{3}(|u><v|+|v><u|, \quad where <u|v>=0$$ In matrix form Homework Equations $$\rho=\sum_i p_i |\psi><\psi|$$ The Attempt at a Solution [/B] The two first factors ##\frac{1}{3}|u><u|##...

Consider the following tree-level Feynman diagrams for the ##W^{+}W^{-} \to W^{+}W^{-}## scattering process. The matrix element for this diagram can be read off from the associated quartic term ##\mathcal{L}_{WWWW}## in the electroweak boson self-interactions, where ##\mathcal{L}_{WWWW} =...

The system in which I tried to calculate the Hamiltonian matrix was a particle in a stadium (Billiard stadium). And I used the principle where we take a rectangle around the stadium in which the parts outside the stadium have a very high potential V0. We know the wave function of a rectangular...

Let ##\Lambda## be a Lorentz transformation. The matrix representing the Lorentz transformation is written as ##\Lambda^\mu{}_\nu##, the first index referring to the rows and the second index referring to columns. The defining relation (necessary and sufficient) for Lorentz transforms is...

im wondering if anyone has any plausible theories that would disprove the possibility of the matrix? ive doe quite a few google searches and all i can find is the concept is quite possible. but surely there must be some ideas that would imply it can't be.

https://uploads.tapatalk-cdn.com/20170308/78feec183e9672f563c5e41b4c52e1d9.jpg https://uploads.tapatalk-cdn.com/20170308/4ad8560adf9e090969c38515a31d1407.jpg Please help, I know the definition of a cosine of a matrix is cos(a) = I-1/2!A^2+1/4!A^4-... But I am unsure how this would help me find...

Homework Statement We assume from ODE theory that given a smooth A: I → gl(n;R) there exists a unique smooth solution F : I → gl(n;R), defined on the same interval I on which A is defined, of the initial value problem F' = FA and F(t0) = F0 ∈ gl(n;R) given.(i) Show that two solutions Fi : I →...

I have some experience with non-linear least squares curve fitting. For instance, if I want to fit a Gaussian curve to a set of data, I would use a non-linear least squares technique. A "model" matrix is implemented and combined with the observed data. The solution is found by applying well...

Homework Statement Please bear with the length of this post, I'm taking it one step at a time starting with i) Let A: I → gl(n, R) be a smooth function where I ⊂ R is an interval and gl(n, R) denotes the vector space of all n × n matrices. (i) If F : I → gl(n, R) satisfies the matrix ODE F'...

Hi, I am studying Matrix chain Multiplication to find out the optimal way of multiplying a series of matrices so that we can reduce the number of multiplications. I have got this example from the book which multiplies the matrices having dimensions given below: A1 30 * 35...

Hi, I'm trying to understand which mathematical actions I need to perform to be able to arrive at the solution shown in the uploaded picture. Thank you.

If I have a Hamiltonian diagonal by blocks (H1 0; 0 H2), where H1 and H2 are square matrices, is the density matrix also diagonal by blocks in the same way?

Hi! I have a question regarding making the transition matrix for the corresponding probabilities. The main problem I feel I have here is figuring out how to represent the probabilities in the question in the transition matrix. Like if something is 7 times more likely than something else.. Any...

What are the coordinates on the 3D Bloch ball of a qubit's mixed state of the form: ##\rho=p_{00}|0\rangle \langle 0|+p_{01}|0\rangle \langle 1|+p_{10}|1\rangle \langle 0|+p_{11}|1\rangle \langle 1|##

Anyone know how to get a vertical line between two columns ? Like here in AMSTeX

This is a somewhat vague question that stems from the entries in a directional cosine matrix and I believe the answer will either be much simpler or much more complicated than I expect. So consider the transformation of an arbitrary vector, v, in ℝ2 from one frame f = {x1 , x2} to a primed...

Hello everyone, I am currently trying to understand how we can use feynman diagrams to estimate the matrix element of a process to be used in fermi's golden rule so that we can estimate decay rates. I am trying to learn by going through solved examples, but I am struggling to follow the logic...

Consider the matrix ##C = \gamma^{0}\gamma^{2}##. It is easy to prove the relations $$C^{2}=1$$ $$C\gamma^{\mu}C = -(\gamma^{\mu})^{T}$$ in the chiral basis of the gamma matrices.1. Do the two identities hold in any arbitrary basis of the gamma matrices? 2. How is ##C## related to the charge...

Homework Statement Given a matrix M={{2,1},{1,2}} then value of cos( (π*M)/6 )Homework EquationsThe Attempt at a Solution Eigen values are π/6 and π/2 and eigen vectors are (π/6,{-1,1}) and (π/2,{1,1}). Diagonalize matrix is {{π/6,0},{0,π/2}} I got same value (√3/2)M

Homework Statement Suppose that ##A^2 = 0##. Show that ##A## is not an invertible matrix Homework EquationsThe Attempt at a Solution We can do a proof by contradiction. Assume that ##A^2 = 0## and that ##A## is invertible. This would imply that ##A=0##, which is to say that A is not...

Homework Statement Show that the set GL(n, R) of invertible matrices forms a group under matrix multiplication. Show the same for the orthogonal group O(n, R) and the special orthogonal group SO(n, R). Homework EquationsThe Attempt at a Solution So I know the properties that define a group are...

Homework Statement Show that every matrix A ∈ O(2, R) is of the form R(α) = cos α − sin α sin α cos α (this is the 2d rotation matrix -- I can't make it in matrix format) or JR(α). Interpret the maps x → R(α)x and x → JR(α)x for x ∈ R 2 Homework EquationsThe Attempt at a Solution So I know...