Read about diagonalization | 17 Discussions | Page 1

  1. N

    I Block Diagonal Matrix and Similarity Transformation

    I am looking at page 2 of this document.https://ocw.mit.edu/courses/chemistry/5-04-principles-of-inorganic-chemistry-ii-fall-2008/lecture-notes/Lecture_3.pdf How is the transformation matrix, ν, obtained? I am familiar with diagonalization of a matrix, M, where D = S-1MS and the columns of S...
  2. L

    A How to numerically diagonalize a Hamiltonian in a subspace?

    I want to exactly diagonalize the following Hamiltonian for ##10## number of sites and ##5## number of spinless fermions $$H = -t\sum_i^{L-1} \big[c_i^\dagger c_{i+1} - c_i c_{i+1}^\dagger\big] + V\sum_i^{L-1} n_i n_{i+1}$$ here ##L## is total number of sites, creation (##c^\dagger##) and...
  3. G

    A Diagonalization of Hubbard Hamiltonian

    Hi guys! I am starting to study Hubbard model with application in DFT and I have some doubts how to solve the Hubbard Hamiltonian. I have the DFT modeled to Hubbard, where the homogeneous Hamiltonian is $$ H = -t\sum_{\langle i,j \rangle}\sigma (\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma} +...
  4. G

    Reduced equation of quadratic forms

    Homework Statement Given the following quadric surfaces: 1. Classify the quadric surface. 2. Find its reduced equation. 3. Find the equation of the axes on which it takes its reduced form. Homework Equations The quadric surfaces are: (1) ##3x^2 + 3y^2 + 3z^2 - 2xz + 2\sqrt{2}(x+z)-2 = 0...
  5. RicardoMP

    I Diagonalization and change of basis

    I have the following matrix given by a basis \left|1\right\rangle and \left|2\right\rangle: \begin{bmatrix} E_0 &-A \\ -A & E_0 \end{bmatrix} Eventually I found the matrix eigenvalues E_I=E_0-A and E_{II}=E_0+A and eigenvectors \left|I\right\rangle = \begin{bmatrix} \frac{1}{\sqrt{2}}\\...
  6. Wrichik Basu

    B Why does a matrix diagonalise in this case?

    Why does a matrix become diagonal when sandwiched between "change of matrices" whose columns are eigenvectors?
  7. S

    I Maybe all reals can be listed?

    Firstly, thanks to everyone who participated in my last thread. It helped a lot! This will be the only other topic I can think of posting in physics forums, because, honestly, I don't know very much. I remember sitting down one time and thinking I was quite brilliant when I started to make a...
  8. B

    Diagonalization of Gigantic Dense Hermitian Matrices

    Hi there, This is a question about numerical analysis used particularly in the computational condensed matter or anywhere where one needs to DIAGONALIZE GIGANTIC DENSE HERMITIAN MATRICES. In order to diagonalize dense Hermitian matrices size of 25k-by-25k and more (e.g. 1e6-by-1e6) it is not...
  9. DeathbyGreen

    I Numerical Diagonalization

    Hi! I'm trying to understand how to diagonalize a Hamiltonian numerically. Basically I have a problem with a Hamiltonian such as H = \frac{1}{2}c^{\dagger}\textbf{H}c where c = (c_1,c_2,...c_N)^T The dimensions of the total Hamiltonian are 2N, because each c_i is a 2 spinor. I need to...
  10. BiGyElLoWhAt

    Moment of inertia tensor calculation and diagonalization

    Homework Statement Not sure if this is advanced, so move it wherever. A certain rigid body may be represented by three point masses: m_1 = 1 at (1,-1,-2) m_2 = 2 at (-1,1,0) m_3 = 1 at (1,1,-2) a) find the moment of inertia tensor b) diagonalize the matrix obtaining the eigenvalues and the...
  11. F

    I Diagonalising a system of differential equations

    Given a system of linear differential equations $$x_{1}'=a_{11}x_{1}+a_{12}x_{2}+\cdots a_{1n}x_{n}\\ x_{2}'=a_{21}x_{1}+a_{22}x_{2}+\cdots a_{2n}x_{n}\\ \ldots\\ x_{n}'= a_{n1}x_{1}+a_{n2}x_{2}+\cdots a_{nn}x_{n}$$ this can be rewritten in the form of a matrix equation...
  12. B

    I Eigen Vectors, Geometric Multiplicities and more...

    My professor states that "A matrix is diagonalizable if and only if the sum of the geometric multiplicities of the eigen values equals the size of the matrix". I have to prove this and proofs are my biggest weakness; but, I understand that geometric multiplicites means the dimensions of the...
  13. G

    Diagonalizing a polynomial of operators (Quantum Mechanics)

    The problem asks for the diagonalization of (a(p^2)+b(x^2))^n, where x and p are position and momentum operators with the commutation relation [x,p]=ihbar. a and b are real on-zero numbers and n is a positive non-zero integer. I know that it is not a good way to use the matrix diagonalization...
  14. kostoglotov

    A reasonable analogy for understanding similar matrices?

    I don't really feel that I understand what it means for two matrices to be similar. Of course, I understand the need to understand ideas on their own terms, and that in math analogies are very much frowned upon. In asking if you know of any "reasonable" analogies for what it means for two...
  15. kostoglotov

    How can e^{Diag Matrix} not be an infinite series?

    So, in a section on applying Eigenvectors to Differential Equations (what a jump in the learning curve), I've encountered e^{At} \vec{u}(0) = \vec{u}(t) as a solution to certain differential equations, if we are considering the trial substitution y = e^{\lambda t} and solving for constant...
  16. kostoglotov

    Matrix with repeated eigenvalues is diagonalizable...?

    MIT OCW 18.06 Intro to Linear Algebra 4th edt Gilbert Strang Ch6.2 - the textbook emphasized that "matrices that have repeated eigenvalues are not diagonalizable". imgur: http://i.imgur.com/Q4pbi33.jpg and imgur: http://i.imgur.com/RSOmS2o.jpg Upon rereading...I do see the possibility...
  17. Angelos K

    LAPACK dgeev: parameter had illegal value

    Mod note: I revised the code below slightly, changing the loop control variable i to either j or k. The reason for this is that the browser mistakes the letter i in brackets for the BBCode italics tag, which causes some array expressions to partially disappear. Hello, I am trying for the first...
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