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.

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  1. NATURE.M

    Matrix Diagonalization & Eigen Decomposition

    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...
  2. A

    Cantors diagonalization argument

    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...
  3. L

    Diagonalization of a hamiltonian for a quantum wire

    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...
  4. M

    Understanding Diagonalization and Eigenvalues in Matrix Transformations

    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...
  5. Fernando Revilla

    MHB PolkaDots 54's question at Yahoo Answers (Diagonalization, conic section))

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  6. Fernando Revilla

    MHB JustCurious's question at Yahoo Answers (Diagonalization)

    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.
  7. T

    Diagonalization of Integral Operators: Challenges and Considerations

    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...
  8. 1

    Using diagonalization to find A^k

    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...
  9. M

    Linear algebra - Diagonalization

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  10. T

    Textbook to help me understand eigenvectors and diagonalization

    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...
  11. M

    Diagonalization of 2D wave equation

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  12. N

    Degrees of freedom - matrix diagonalization

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  13. stripes

    Diagonalization and similar matrices

    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...
  14. L

    Cantor's Diagonalization Proof of the uncountability of the real numbers

    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...
  15. J

    Applying Cantor's diagonalization technique to sequences of functions

    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...
  16. S

    Linear Algebra - Diagonalization question

    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...
  17. H

    Linear Algebra proof, diagonalization

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  18. P

    Using diagonalization, prove the matrix equals it's square

    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...
  19. K

    Square Root of a 2x2 Matrix (by diagonalization)

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  20. U

    Diagonalization of square matrix if not all eigenvalues are distinct of

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

    Diagonalization with nilpotent matrices

    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...
  22. H

    Diagonalization of Eigenvalues: A Mistake in Homework Answer?

    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...
  23. Y

    Diagonalization of symmetric bilinear function

    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...
  24. G

    Simutanious diagonalization of 2 matrices

    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...
  25. S

    Finding a Matrix P for Non-symmetric Diagonalization: To Normalize or Not?

    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...
  26. M

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  27. P

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  28. E

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  29. S

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  30. B

    Order of Eigenvectors in Diagonalization

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  31. G

    Eigenvector orthogonality and unitary operator diagonalization

    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...
  32. A

    Diagonalization of Specif Matrix

    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...
  33. F

    The question about diagonalization

    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...
  34. T

    Diagonalization of an almost diagonal matrix

    Homework Statement If we have a n x n matrix with 1 on the diagonal entries apart from the ith column which has a -1. As well as this ith row can have any real number in each entry. Other than this the matrix is 0 everywhere. Show this matrix is diagonalisable. Homework Equations...
  35. Y

    Simultaneous diagonalization of two hermitian operators

    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...
  36. B

    Diagonalization of Matrices: Confusion about Eigenvalues and Eigenvectors

    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...
  37. TheFerruccio

    Finding eigenbasis and diagonalization

    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...
  38. S

    Linear algebra Applications of Diagonalization

    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?
  39. D

    Stress-energy tensor diagonalization

    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)...
  40. X

    Diagonalizing Matrix A: Eigenvalues, Eigenvectors, Matrix P & D

    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...
  41. E

    Eigenvalues/vectors diagonalization

    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
  42. B

    Use Cantor's Diagonalization on the set of Natural Numbers?

    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...
  43. M

    Can Two Hermitian Matrices be Simultaneously Diagonalized if They Commute?

    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...
  44. I

    Finding eigenvectors for diagonalization

    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...
  45. F

    Diagonalization, which eigenvector is found?

    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...
  46. S

    Show that this orthogonal diagonalization is a singular value decomposition.

    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...
  47. O

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    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...
  48. J

    Diagonalization of large non-sparse matrices

    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...
  49. A

    Simultaneous diagonalization and replacement of operators with eigenvalues ?

    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...
  50. K

    What are the criteria for determining if a matrix is diagonalizable?

    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...
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