What is Diagonalization: Definition and 131 Discussions

In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.
The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, which appeared in 1874.
However, it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Gödel's incompleteness theorems and Turing's answer to the Entscheidungsproblem. Diagonalization arguments are often also the source of contradictions like Russell's paradox and Richard's paradox.

View More On Wikipedia.org
  1. K

    I Change of basis matrix for point group C3V

    I am looking at the point group C<sub>3v</sub> described shown here. I am trying to understand the block diagonalization process. The note says that changing the basis in the following way will result in the block diagonal form. What is the rationale for choosing the new basis. Is it...
  2. nomadreid

    I Spectral theorem for Hermitian matrices-- special cases

    I have a proof in front of me that shows that for a normal matrix M, the spectral decomposition exists with M=PDP-1 where P is an invertible matrix and D a matrix that can be represented by the sum over the dimension of the matrix of the eigenvalues times the outer products of the corresponding...
  3. L

    A Can a Non-Diagonal Hermitian Matrix be Diagonalized Using Unitary Matrix?

    Every hermitian matrix is unitary diagonalizable. My question is it possible in some particular case to take hermitian matrix ##A## that is not diagonal and diagonalize it UAU=D but if ##U## is not matrix that consists of eigenvectors of matrix ##A##. ##D## is diagonal matrix.
  4. C

    Diagonalizing a Matrix: Understanding the Process and Power of Matrices

    For this, Dose someone please know where they get P and D from? Also for ##M^k##, why did they only raise the the 2nd matrix to the power of k? Many thanks!
  5. P

    Question about hollow matrix and diagonalization

    A quick and simple question. I just realized that this has been posted in the wrong section, but ill give it a try anyway. Does anyone know if it's possible to diagonalize a hollow matrix? What i mean by a hollow matrix is a matrix with only zero entries along the diagonal.
  6. D

    Diagonalizing of Hamiltonian of electron and positron system

    What I did was first noting that ##\hat{\vec{S}}_1\cdot\hat{\vec{S}}_2=\frac{1}{2}(\hat{\vec{S}}^2-\hat{\vec{S}}_1^2-\hat{\vec{S}}_2^2)##, but these operators don't commute with ##\hat{S}_{1_z}## and ##\hat{S}_{2_z}##, this non the decoupled basis ##\ket{s_1,s_2;m_1,m_2}## nor the coupled one...
  7. BWV

    I Cantor's diagonalization on the rationals

    Question that occurred to me, most applications of Cantors Diagonalization to Q would lead to the diagonal algorithm creating an irrational number so not part of Q and no problem. However, it should be possible to order Q so that each number in the diagonal is a sequential integer- say 0 to...
  8. A

    I Canonical Form for quadratic equations *with* linear terms

    Hello: I'm not sure if there's an accepted canonical form for a quadratic equation in two (or more) variables: $$ax^2+by^2+cxy+dx+ey+f=0$$ Is it the following form? (using the orthogonal matrix Q that diagonalizes the quadratic part): $$ w^TDw+[d \ \ e]w+f=0$$ $$w^TDw+Lw+f=0$$ where $$...
  9. S

    Diagonalizing a matrix given the eigenvalues

    The following matrix is given. Since the diagonal matrix can be written as C= PDP^-1, I need to determine P, D, and P^-1. The answer sheet reads that the diagonal matrix D is as follows: I understand that a diagonal matrix contains the eigenvalues in its diagonal orientation and that there must...
  10. A

    MHB Cantor Diagonalization | Find a Number Not on the List

    Consider the following list of numbers. Using Cantor's diagonalization argument, find a number not on the list (use 2 and 4 when applying Cantor' argument). Give a brief explanation of the process. 0.123456876… 0.254896487… 0.143256876… 0.758468126… 0.534157162…
  11. M

    A Diagonalization of 2x2 Hermitian matrices using Wigner D-Matrix

    Motivation: Due to the spectral theorem a complex square matrix ##H\in \mathbb{C}^{n\times n}## is diagonalizable by a unitary matrix iff ##H## is normal (##H^\dagger H=HH^\dagger##). If H is Hermitian (##H^\dagger=H##) it follows that it is also normal and can hence be diagonalized by a...
  12. C

    Doubt about Jacobi's diagonalization method

    Good Morning, I am using the Jacobi diagonalization method for symmetric matrices and I have realized that as the number of iterations progresses, the speed with which the larger element (in absolute value) outside the diagonal of the diagonal becomes smaller Matrices are increasing (graphical...
  13. N

    A Block Diagonalization - Representation Theory

    How does one go about finding a matrix, U, such that U-1D(g)U produces a block diagonal matrix for all g in G? For example, I am trying to figure out how the matrix (7) on page 4 of this document is obtained.
  14. 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...
  15. 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...
  16. Jarvis323

    I Understanding Cantor's Diagonalization Proof: A Brief Explanation

    I wrote a long response hoping to get to the root of AlienRender's confusion, but the thread closed before I posted it. So I'm putting it here. It seams you're argument is something like this: let ##f(n) = 2n##, this is a one-one and onto mapping of the natural numbers onto the even natural...
  17. 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} +...
  18. 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 ##...
  19. LarryC

    Simultaneous Diagonalization for Two Self-Adjoint Operators

    (a) and (b) are fairly traditional, but I have trouble understanding the phrasing of (c). What makes the infinite dimensionality in (c) different from (a) and (b)?
  20. L

    A Hermitian Operators and Projectors in Linear Algebra

    Matrix \left[ \begin{array}{rr} 1 & 1 \\ 0& 0 \\ \end{array} \right] is not symmetric. When we find eigenvalues of that matrix we get ##\lambda_1=1##, ##\lambda_2=0##, or we get matrix \left[ \begin{array}{rr} 1 & 0 \\ 0& 0 \\ \end{array} \right]. First matrix is not hermitian, whereas second...
  21. L

    Why Is My Matrix Not Diagonal After Transformation?

    Homework Statement Form unitary matrix from eigen vectors of ##\sigma_y## and using that unitary matrix diagonalize ##\sigma_y##. \sigma_y= \begin{bmatrix} 0 & -i & \\ i & 0 & \\ \end{bmatrix}[/B]Homework Equations Eigen vectors of ##\sigma_y## are...
  22. 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}}\\...
  23. Luck0

    A Diagonalization of adjoint representation of a Lie Group

    So, we know that if g is a Lie algebra, we can take the cartan subalgebra h ⊂ g and diagonalize the adjoint representation of h, ad(h). This generates the Cartan-Weyl basis for g. Now, let G be the Lie group with Lie algebra g. Is there a way to diagonalize the adjoint representation Ad(T) of...
  24. 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?
  25. L

    A Hubbard model diagonalization in 1D K-space for spinless Fermions

    I am trying to diagonalize hubbard model in real and K-space for spinless fermions. Hubbard model in real space is given as: H=-t\sum_{<i,j>}(c_i^\dagger c_j+h.c.)+U\sum (n_i n_j) I solved this Hamiltonian using MATLAB. It was quite simple. t and U are hopping and interaction potentials. c...
  26. 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...
  27. 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...
  28. DeathbyGreen

    I How can I find the unitary matrix for diagonalizing a Hamiltonian numerically?

    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...
  29. arupel

    A Godel and Diagonalization?

    I can readily accept that the Godel sentence The theorem is that "This theorem is not provable" can be expressed in the language of Peanno Arithmetic. 2. Godel on the other side of a correspondence with the above, first translates the Godel sentence using the Godel numbering system 3. Having...
  30. 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...
  31. 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...
  32. 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...
  33. 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...
  34. Dewgale

    Diagonalization and Unitary Matrices

    \Homework Statement Let B = ## \left[ \begin{array}{ccc} -1 & i & 1 \\ -i & 0 & 0 \\ 1 & 0 & 0 \end{array} \right] ##. Find a Unitary transformation to diagonalize B. Homework Equations N/A The Attempt at a Solution I have found both the Eigenvalues (0, 2, -1) and the Eigenvectors, which are...
  35. S

    Diagonalization in R: Can Matrix Be Diagonalized?

    Homework Statement Is the following matrix diagonalizable in R? [ 2 1 0 ] [ 1 3 -1 ] [ -1 2 3 ] Homework EquationsThe Attempt at a Solution I've checked my work and found one eigenvalue = 2, with the corresponding eigenvector = [1, 0, 1]. My question is -- Because I have one eigenvector, can...
  36. 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...
  37. D

    Diagonalizing a Matrix: Steps and Verification

    Homework Statement Diagonalize matrix using only row/column switching; multiplying row/column by a scalar; adding a row/column, multiplied by some polynomial, to another row/column. Homework EquationsThe Attempt at a Solution After diagonalization I get a diagonal matrix that looks like...
  38. 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...
  39. 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...
  40. diracdelta

    Quadratic form and diagonalization

    Homework Statement Find diagonal shape of next quadratic form ( using eigenvalues and eigenvectors) Q(x,y)= 5x2 + 2y2 + 4xy. What is curve { (x,y)∈ ℝ| Q(x,y)= λ1λ2, where λ1 and λ2 are eigenvalues of simetric matrix joined to quadratic form Q. Draw given curve in plane. The Attempt at a...
  41. 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...
  42. D

    Eigenvalues and diagonalization of a matrix

    When you diagonalize a matrix the diagonal elements are the eigenvalues but how do you know which order to put the eigenvalues in the diagonal elements as different orders give different matrices ? Thanks
  43. Qturtle

    Block diagonalization of a matrix

    Hi. i have a 4x4 matrix \begin{pmatrix} 0 & 1 & 1 & 1\\ 1 & 0 & i & -i\\ 1 & -i & 0 & i\\ 1 & i & -i & 0\\ \end{pmatrix} it has 2 eigenvalues and i want to block diagonalize it into a 2x2 block diagonal matrix. i can't seem to find the proper way to do that. do i need to have a commuting matrix...
  44. R

    Impose Uniqueness on Diagonalization of Inertia Tensor?

    Given an inertia tensor of a rigid body I, one can always find a rotation that diagonalizes I as I = RT I0 R (let's say none of the value of the inertia in I0 equal each other, though). R is not unique, however, as one can always rotate 180 degrees about a principal axis, or rearrange the...
  45. M

    Minimal polynomial and diagonalization of a block matrix

    Homework Statement . Let ##X:=\{A \in \mathbb C^{n\times n} : rank(A)=1\}##. Determine a representative for each equivalence class, for the equivalence relation "similarity" in ##X##. The attempt at a solution. I am a pretty lost with this problem: I know that, thinking in terms of...
  46. D

    Comp Sci Diagonalizing Matrices in C++: A Beginner's Guide

    Homework Statement Hi :) I want to write a program in c++ to diagonalize given matrix. However, I'm stuck and I don't have any ideas to do it. I found a linear algebra library for c++ but I could not solve my problem because I don't know how to solve a 2nd order equation. Can you help me...
  47. T

    Diagonalization of 8x8 matrix with Euler angles

    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}...
  48. H

    Diagonalization of a Hamiltonian for two fermions

    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...
  49. A

    Exact diagonalization by Bogoliubov transformation

    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 &...
  50. A

    Hamiltonian diagonalization

    Homework Statement The exercise: https://www.physicsforums.com/attachment.php?attachmentid=64229&d=1385257430 Homework Equations Are in my attempt at a solution. I am sure it would be easier to use the transformation equation for the operator and plug it into the diagonalized Hamiltonian...