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. J

    Diagonalization 2: Explaining Theorem 2.5

    I am reading yet another theorem and was wondering If I could get more clarification on it. Theorem 5.2: Let A be in M_n_x_n(F). Then a scalar \lambda is an eigenvalue of A if and only if det(A - \lambdaI_n) = 0. Proof: A scalar \lambda is an eigenvalue of A if and only if there exists a...
  2. M

    Finding an Invertible Matrix for Matrix Diagonalization

    Homework Statement A = -10 6 3 -26 16 8 16 -10 -5 B = 0 -6 -16 0 17 45 0 -6 -16 (a) Show that 0, -1 and 2 are eigenvalues both of A and of B . (b) Find invertible matrices P and Q so that (P^-1)*(A)*(P) = (Q^-1)*(B)*(Q)= 0 0 0 0 -1 0 0 0 2 (c) Find an invertible...
  3. B

    Linear algebra diagonalization

    linear algebra diagonalization :( Homework Statement determine whether the given matrix is diagonalizable. where possible, find a matrix S such that S-1AS=Diag(λ1+λ2,...,λn) Homework Equations The Attempt at a Solution I was able to find the eigenvalues, which are λ=1,-4,4. This is given in...
  4. Deneb Cyg

    Square Matrix A that is not Diagonalizable but A^2 is Diagonalizable

    The question is "give an example of a square matrix A such that A^2 is diagonalizable but A is not." I know that if A^2 is diagonalizable, A^2 = P(D^2)P^-1. And if A is not diagonalizable, there is no invertible matrix P and diagonal matrix D such that A=PDP^-1. However I'm not sure how...
  5. C

    Matrix Diagonalization

    Homework Statement My linear algebra textbook defines... similar matrices: A = C^-1BC diagonalized similar matrices: A = CDC^-1 A^n = C^-1*D^n*C Why do the C^-1 and C's get switched around between the definitions? Doesn't order of multiplication matter? Are these the correct...
  6. A

    Cantor diagonalization argument

    sorry for starting yet another one of these threads :p As far as I know, cantor's diagonal argument merely says- if you have a list of n real numbers, then you can always find a real number not belonging to the list. But this just means that you can't set up a 1-1 between the reals, and...
  7. J

    Bloch's theorem and diagonalization of translation operator

    I'm now interested in a Schrödinger's equation \Big(-\frac{\hbar^2}{2m}\partial_x^2 + V(x)\Big)\psi(x) = E\psi(x) where V does not contain infinities, and satisfies V(x+R)=V(x) with some R. I have almost already understood the Bloch's theorem! But I still have some little problems...
  8. S

    Quadratic forms, diagonalization

    Can a quadratic form always be diagonalised by a rotation?? Thx in advance
  9. J

    Matrix, eigenvalues and diagonalization

    Matrix A= 1 2 0 2 1 0 2 -1 3 i got eigenvalues k=3 k=-1 what do i do after that to prove it is not able to be diagonalized
  10. B

    What is wrong with this diagonalization problem.

    Homework Statement A is [4 0 1 2 3 2 1 0 4] Find an invertible P and a diagonal D so that D=P-1AP. I keep getting two linearly dependent eigenvalues which means it's not diagonal but this problem doesn't state "If it can't be done explain why" or anything like that. So I just...
  11. F

    Proving the Diagonalizability of a Real 2x2 Matrix Using Invertible Matrices

    Homework Statement Let A be a 2x2 real matrix which cannot be diagonalized by any matrix P (real or complex). Prove there is an invertible real 2x2 matrix P such that P^{-1}AP = \left( \begin{array}{cc} \lambda & 1 \\ 0 & \lambda \end{array} \right) I know how to diagonalize a matrix...
  12. J

    Linear Algebra: Diagonalization, Transpose, and Disctinct Eigenvectors.

    Homework Statement Show that if an nxn matrix A has n linearly independent eigenvectors, then so does A^T The Attempt at a Solution Well, I understand the following: (1) A is diagonalizable. (2) A = PDP^-1, where P has columns of the independent eigenvectors (3) A is...
  13. J

    Regarding Diagonalization of Matrix by Spectral Theorem

    According to the spectral theorem for self-adjoint operators you can find a matrix P such that P^{-1}AP is diagonal, i.e. P^{T}AP (P can be shown to be orthogonal). I'm not sure what the result is if the same can be done for the following square (size n X n) and symmetric matrix of the form: A=...
  14. A

    Orthogonal Diagonalization of a Symmetric Matrix

    Homework Statement Orthogonally diagonalize the matrix: | 2 1 1| | 1 2 1| | 1 1 2| Homework Equations Since this only has...
  15. A

    Diagonalization, eigenvectors, eigenvalues

    [SOLVED] diagonalization, eigenvectors, eigenvalues Homework Statement Find a nonsingular matrix P such that (P^-1)*A*P is diagonal | 1 2 3 | | 0 1 0 | | 2 1 2 | Homework Equations I am at a loss on how to do this. I've tried finding the eigen values but its getting me...
  16. B

    Diagonalization & Eigen vectors proofs

    Homework Statement Question 1: A) Show that if A is diagonalizable then A^{T} is also diagonalizable. The Attempt at a Solution We know that A is diagonalizable if it's similar to a diagonal matrix. So A=PDP^{-1} A^{T}=(PDP^{-1})^{T} which gives A^{T}=(P^{-1})^{T}DP^{T} as...
  17. B

    Diagonalization and Matrix similarity

    Question1 : a) Show that if A is nonsingular and diagonalizable then A^-1 is diagonalizable b) Show that if A is diagonalizable then A^T is diagonalizable Question 2 Show that if A is similar to B and B similar to C, then A is similar to C.
  18. C

    How exactly does diagonalization work and how is it useful in qm?

    how exactly does it work and how is it useful in qm?
  19. N

    Diagonalization Algorithm for Large Matrices: Any Suggestions?

    Does anyone here know of any fast algorithms to diagonize large, symmetric matrices, that are mostly zeros? (by large I mean 300x300 up to several million by several million)
  20. E

    Diff equation by diagonalization

    Homework Statement solve initial value problem for the equation dx/dt = Ax where A = [1 -1] [0 1] x(0) = [1, 1]T x(t) = S*elambda*t*S -1*x(0) where S is diagonal matrix, lambda is eigenvalue; The Attempt at a Solution I tried to diagonalize it, but I get one eigenvalue =1 mult 2 and I don't...
  21. E

    Symmetric matrix and diagonalization

    This is a T/F question: all symmetric matrices are diagonalizable. I want to say no, but I do not know how exactly to show that... all I know is that to be diagonalizable, matrix should have enough eigenvectors, but does multiplicity of eigenvalues matter, i.e. can I say that if eignvalue...
  22. H

    Geometric and Physical Interpretation of Diagonalization

    I am a second year student in quantum mechanics. I heard in lecture that simultaneous diagonalization of matrices is important in quantum mechanics. I would like to know why is it significant when two matrices can be simultaneous diagonalized, and what is the geometric and physical...
  23. B

    Diagonalization of a matrix with repeated eigenvalues

    Hey guys, I know its possible to diagonalize a matrix that has repeated eigenvalues, but how is it done? Do you simply just have two identical eigenvectors?? Cheers Brent
  24. E

    Is the set of infinite strings of {a,b} countable?

    I am stumpt on this problem: Use Cantor's diagonalization method to show that the set of all infinite strings of the letters {a,b} is not countable: Can I have a hint? :redface:
  25. L

    Basic metric diagonalization questions

    I understand it is always possible to diagonalize a metric to the form diag[1,-1,\dots,-1] at any given point in spacetime because the metric is symmetric and we can always re-scale our eigenvectors. But is this achievable via a coordinate transformation? That is, would the basis...
  26. S

    What is Special Diagonalization?

    i need to know about specially diagonalization, is it orthogonal diagonalization or something different?
  27. B

    Simultaneous Diagonalization

    I am having a hard with parts of this problem: Suppose V is a 3-dimensional complex inner-product space. Let B1 = {|v1>,|v2>,|v3>} be an orthonormal basis for V. Let H1 and H2 be self-adjoint operators represented in the basis B1 by the Hermitian matrices. I won't list them, but they...
  28. M

    How Do I Calculate the Inverse of Matrix P and Determine Eigenvectors?

    Could someone tell me how to get the P(inverse), P^-1. For example I read all the examples in my book and it has like given the matrix for P and then it finds the matrix for P(inverse)Vo , how do i find P^-1. Plz help me quickly. Ex. P= | 2 -1 | asdfasf| 3 as1 | and Vo=| 1 |...
  29. O

    A new point of view on Cantor's diagonalization arguments

    Hi, In this pdf (+ its links)http://www.geocities.com/complementarytheory/NewDiagonalView.pdf you can find a new point of view on Cantor's diagonalization arguments. I really want to send a BIG THANK YOU to Matt grime and Hurkyl for their hard time with me. Yours, Organic
  30. O

    Cantor's Diagonalization function on the combinations list

    PLEASE READ THIS POST UNTIL ITS LAST WORD, BEFORE YOU REPLY. THANK YOU. Let us check these lists. P(2) = {{},{0},{1},{0,1}} = 2^2 = 4 and also can be represented as: 00 01 10 11 P(3) = {{},{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}} = 2^3 = 8 and also can be represented...
  31. O

    A question on Cantor's second diagonalization argument

    Hi, Cantor used 2 diagonalization arguments. On the first argument he showed that |N|=|Q|. On the second argument he showed that |Q|<|R|. I have some question on the second argument. From Wikipedia, the free encyclopedia: http://en.wikipedia.org/wiki/Cantor's_diagonal_argument...
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