Diagonalization Definition and 131 Threads

I am trying to diagonalize the following matrix: M = \left( \begin{array}{cccc} 0 & 0 & 0 & a \\ 0 & 0 & -a & 0 \\ 0 & -a & 0 & -A \\ a & 0 & -A & 0 \end{array} \right) a and A are themselves 2x2 symmetric matrices: a = \left( \begin{array}{cc} a_{11} & a_{12}\\ a_{12} & a_{22}...

Homework Statement Hi, I want to diagonalize the Hamiltonian: Homework Equations H=\phi a^{\dagger}b + \phi^{*} b^{\dagger}a a and b are fermionic annihilation operators and \phi is some complex number. The Attempt at a Solution Should I use bogoliubov tranformations? I...

Hello all, I am developing a model of multiple gaps in a square lattice. I simplified the associated Hamiltonian to make it quadratic. In this approximation it is given by, H = \begin{pmatrix} \xi_\mathbf{k} & -\sigma U_1 & -U_2 & -U_2\\ -\sigma U_1 & \xi_{\mathbf{k}+(\pi,\pi)} & 0 &...

Do these terms practically refer to the same thing? Like a matrix is diagonalizable iff it can be expressed in the form A=PDP^{-1}, where A is n×n matrix, P is an invertible n×n matrix, and D is a diagonal matrix Now, this relationship between the eigenvalues/eigenvectors is sometimes...

I am sure you are all familiar with this. The number generated by picking different integers along the diagonal is different from all other numbers previously on the list. But you could just put this number as next element on the list. Of course that just creates a new number which is missed...

I try diagonalize the Hamiltonian for a 1D wire with proximity-induced superconductivity. In the case without a superconductor is all fine. However, with a superconductor I don't get the correct result for the energy spectrum of the Hamiltonian (arxiv:1302.5433) H=\eta(k)τz+Bσ_x+αkσ_yτ_z+Δτ_x...

Let's say I have a matrix M such that for vectors R and r in xy-coordinate system: R=Mr Suppose we diagonalized it so that there is another matrix D such that for vectors R' (which is also R) and r' (which is also r) in x'y'-coordinate system: R'=Dr' D is a matrix with zero elements except for...

Here is the question: Here is a link to the question: Diagonalization to identify conic section? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Here is the question: Here is a link to the question: Diagnalization with matrices? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

So, obviously one can diagonalize any self-adjoint transformation on a finite dimensional vector space. This is pretty simple to prove. What I'm curious about is integral operators. How does this proof need to be adapted to handle integral operators? What goes wrong? What do we need to account...

Homework Statement A = \begin{pmatrix} 1 & 4\\ 2 & -1 \end{pmatrix} Find A^n and A^{-n} where n is a positive integer. Homework Equations The Attempt at a Solution (xI - A) = \begin{pmatrix} x-1 & -4\\ -2 & x+1 \end{pmatrix} det(xI - A) = (x-3)(x+3) λ_1 = 3\quad...

does someone know how to solve the following? Homework Statement "Find all matrices A so I A 0 I is Diagonalizable " this is a picture of the matrice http://i46.tinypic.com/258v514.jpg How can I find A?

Hi, I'm currently self-teaching myself some mathematics needed to study physics. I'm working through the book Mathematical Methods in the Physical Sciences by Mary L Boas. The book is a well known one, and it's used in many physics programs to teach their math courses. However, I've read the...

Homework Statement I've just derived the 1D wave equation for a continuous 1D medium from a classical Hamiltonian. I simply wrote Hamilton's equations, where the derivatives here must be functional derivatives (e.g. δ/δu(x)) since p and u are functions of x, and I got the wave equation (see...

So when dealing with a linear transformation, after we have computed the matrix of the linear transformation, and we are asked "is this matrix diagonalizable", I begin by finding the eigenvalues and eigenvectors using the characteristic equation. Once I have found eigenvectors, if I see these...

I have a problem with Cantor's Diagonalization proof of the uncountability of the real numbers. His proof appears to be grossly flawed to me. I don't understand how it proves anything. Please take a moment to see what I'm talking about. Here is a totally abstract pictorial that attempts...

As usually, I type the problem and my attempt at the solution in LaTeX. Ok, so for the last part (c), I obviously have the diagram down, now I just have to construct the nested sequence of functions that converges at every point in A. I drew a diagram to help illustrate the idea...

Homework Statement Suppose A = SΛS^{-1}. What is the eigenvalue matrix for A + 2I? What is the eigenvector matrix? Check that A + 2I = ()()()^{-1}. The Attempt at a Solution I think I'm pretty close I'm just not sure what to do next: A + 2I = SΛS^{-1} + 2I = SΛS^{-1} + 2SS^{-1} ? now...

Homework Statement Let N be a 2x2 matrix such that N^2 = 0. Prove that either N = 0 or N is similar to the Matrix ((0,0),(1,0)) Homework Equations N/A The Attempt at a Solution N^2=0 Assume N ≠0 Show N is similar to ((0,0),(1,0)) Need to find a basis of R^2 {V_1,V_2}...

Homework Statement Suppose that A is a 2x2 matrix with eigenvalues 0 and 1. Using diagonalization, show that A2 = A The Attempt at a Solution Let A=\begin{pmatrix}a&b\\c&d\end{pmatrix} Av=λv where v=\begin{pmatrix}x\\y\end{pmatrix} and x,y≠0 If λ=0 then ax+by=0 and cx+dy=0 If λ=1...

Is it possible to diagonalize such matrix? and how would one do it?

So my professor gave me an extra problem for Linear Algebra and I can't find anything about it in his lecture notes or textbooks or online. I think I've made it through some of the more difficult stuff, but I am running into a catch at the end. Homework Statement Find [;T(p(x))^{500};] when...

Homework Statement I think my teacher made a mistake in his homework answer. I need to verify this for practice. The answer I got is below. The answer the teacher has is in the pdf. Homework Equations Please refer to attached pdf The Attempt at a Solution So there is two...

According to duality principle, a bilinear function \theta:V\times V \rightarrow R is equivalent to a linear mapping from V to its dual space V*, which can in turn be represented as a matrix T such that T(i,j)=\theta(\alpha_i,\alpha_j). And this matrix T is diagonalizable, i.e...

Homework Statement From Principles of Quantum Mechanics, 2nd edition by R Shankar, problem 1.8.10: By considering the commutator, show that the following Hermitian matrices may be simultaneously diagonalized. Find the eigenvectors common to both and verify that under a unitary transformation...

Homework Statement Find a matrix P such that P^{-1}AP is diagonal and evaluate P^{-1}AP. A= [2 5] [2 3] The Attempt at a Solution First off, I Found the Eigenvalues, which turned out to be: \lambda = \frac{5 \pm \sqrt{41}}{2} This gave me the two Eigenvectors...

Hi, Can anyone help me prove that two commuting matrices can be simultaneously diagonalized? I can prove the case where all the eigenvalues are distinct but I'm stumped when it comes to repeated eigenvalues. I came across this proof online but I am not sure how B'_{ab}=0 implies that B is...

Is every complex symmetric (NOT unitary) matrix M diagonalizable in the form U^T M U, where U is a unitary matrix? Why?

1. Is its possible diagonalization of metric matrix (g_{uv}) in general relativity? 2. If we include imaginary numbers, can this help?

Nevermind -- Polygons and Polywags.

I essentially know how to find eigenvalues and thus eigenvectors, though when solving a problem about diagonalization I do not know how to order them (as in, I can find all the eigenvectors but do not know which order to place them into find my X that diagonalizes my A) In the examples of my...

Homework Statement For reference: Problem 1.8.5 parts (3) , R. Shankar, Principles of Quantum Mechanics. Given array \Omega , compute the eigenvalues ( e^i^\theta and e^-^i^\theta ). Then (3) compute the eigenvectors and show that they are orthogonal. Homework Equations Eulers...

Now this could seem like a homework problem...but it's not. (I guess you'd need to believe me or just choose not to answer my question.) I'm trying to compute the eigenvalues of a matrix and it's a little more irritating than I'd expected. All I really care is if they're positive (so all I...

I have the confusion that the one question is shown below: Consider the following matrix: A= [1 -1;1 1] which is 2x2 matrix, the column of that is [1 1] and [-1 1] respectively. What happens when we apply A to vector v a large number of times? Hoping someone can help me solve this...

I decided to go over the mathematical introductions of QM again.The text I use is Shankar quantum, and I came across this theorem: "If \Omega and \Lambda are two commuting hermitain operators, there exists (at least) a basis of common eigenvectors that diagonalizes them both." in the proof...

Homework Statement Ok so I have to construct a real symmetric matrix R whose eigenvalues are 2,1,-2 and who corresponding normalized eigenvectors are bla bla bla.. So let the matrix of eigenvalues down diagonal be E and matrix of eigen vectors be V Is R = VEV^T or R = V^TEV?? How...

Homework Statement Find the eigenbasis and diagonalize. Homework Equations \mathbf{A} = \left[ {\begin{array}{ccc} 5& \frac{8}{3} & \frac{-2}{3} \\ 2 & \frac{2}{3}& \frac{4}{3} \\ -4 & \frac{-4}{3} & \frac{-8}{3}\\ \end{array} } \right] The Attempt at a Solution I find the characteristic...

Homework Statement I attached the problem as an image, its easier to see this way. Homework Equations The Attempt at a Solution I understand how to find diagonal matricies using eigenvalues but I'm lost on the Y part. How do I find the vector Y?

This question probably applies to symmetric rank-2 tensors in general, but I've been thinking about it specifically in the context of the stress-energy tensor. For any stress-energy tensor and any metric (with signature -, +, +, +), is it possible to find a coordinate transformation that a)...

Homework Statement A=\left[\begin{array}{ccc}1 & 0 & 0\\ 0 & 1 & -1\\ 0 & 0 & 2\end{array} a) Find the eigenvalues and corresponding eigenvectors of matrix A. b)Find the matrix P that diagonalizes A. c)Find the diagonal matrix D suh that A = PDP-1, and verify the equality. d) Find...

Homework Statement Suppose that A \in Mnxn(F) has two distinct eigenvalues \lambda_{1} and \lambda_{2} and that dim(E_{\lambda_{1}}) = n - 1. Prove that A is diagonalizable Homework Equations The Attempt at a Solution hmm, I'm not sure.. how would I start this? thanks

Homework Statement This is actually only related to a problem given to me but I still would like to know the answer. From my understanding, Cantor's Diagonalization works on the set of real numbers, (0,1), because each number in the set can be represented as a decimal expansion with an...

hello, i am having some trouble understanding simultaneous diagonalization. i have understood the proof which tells us that two hermitian matrices can be simultaneously diagonalized by the same basis vectors if the two matrices commute. but my book then shows a proof for the case when the...

Homework Statement Let A = \left[ \begin{array}{cc} -6 & 0.25 \\ 7 & -3 \end{array} \right] Find an invertible S and a diagonal D such that S^{-1}AS=DHomework Equations ...The Attempt at a Solution So first I need to get eigenvalues so I can get the eigenvectors which will give me the...

Hi! This might be a silly question, but I can't seem to figure it out and have not found any remarks on it in the literature. When diagonalizing an NxN matrix A, we solve the characteristic equation: Det(A - mI) = 0 which gives us the N eigenvalues m. Then, to find the eigenvectors v...

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'm trying to solve the following problem (not homework :smile:) which is a strange form of diagonalization problem. Standard references and papers didn't turn up anything for me. Does anyone see possible approach for this? - Given n x n full rank random matrices A1, A2, ... A9 Find length...

Dear physics friends: I am using a Potts model to study protein folding. In short, the partition function of the problem is written as the sum of the eigenvalues of the transfer matrix each to the Nth power (the transfer matrix factors the expression exp(H/Temperature) where H is the...

Apparently, if I have a Hamiltonian that contains an operator, and that operator commutes with the Hamiltonian, not only can we "simultaneously diagonalize" the Hamiltonian and the operator, but I can go through the Hamiltonian and replace the operator with its eigenvalue everywhere I see it...

Homework Statement 1) Let's say I was trying to find the eigenvalues of a matrix and came up with the following characteristic polynomial: λ(λ-5)(λ+2) This would yield λ=0,5,-2 as eigenvalues. I'm kinda thrown off as to what the algebraic multiplicity of the eigenvalue 0 would be? I'm pretty...