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.
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...
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...
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.
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!
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.
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...
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...
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
$$...
The following matrix is given.
Since the diagonal matrix can be written as C= PDP^-1, I need to determine P, D, and P^-1.
The answer sheet reads that the diagonal matrix D is as follows:
I understand that a diagonal matrix contains the eigenvalues in its diagonal orientation and that there must...
Consider the following list of numbers. Using Cantor's diagonalization argument, find a number not on the list (use 2 and 4 when applying Cantor' argument). Give a brief explanation of the process.
0.123456876…
0.254896487…
0.143256876…
0.758468126…
0.534157162…
Motivation:
Due to the spectral theorem a complex square matrix ##H\in \mathbb{C}^{n\times n}## is diagonalizable by a unitary matrix iff ##H## is normal (##H^\dagger H=HH^\dagger##). If H is Hermitian (##H^\dagger=H##) it follows that it is also normal and can hence be diagonalized by a...
Good Morning,
I am using the Jacobi diagonalization method for symmetric matrices and I have realized that as the number of iterations progresses, the speed with which the larger element (in absolute value) outside the diagonal of the diagonal becomes smaller Matrices are increasing (graphical...
How does one go about finding a matrix, U, such that U-1D(g)U produces a block diagonal matrix for all g in G? For example, I am trying to figure out how the matrix (7) on page 4 of this document is obtained.
I am looking at page 2 of this document.https://ocw.mit.edu/courses/chemistry/5-04-principles-of-inorganic-chemistry-ii-fall-2008/lecture-notes/Lecture_3.pdf
How is the transformation matrix, ν, obtained? I am familiar with diagonalization of a matrix, M, where D = S-1MS and the columns of S...
I want to exactly diagonalize the following Hamiltonian for ##10## number of sites and ##5## number of spinless fermions
$$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}$$
here ##L## is total number of sites, creation (##c^\dagger##) and...
I wrote a long response hoping to get to the root of AlienRender's confusion, but the thread closed before I posted it. So I'm putting it here.
It seams you're argument is something like this:
let ##f(n) = 2n##, this is a one-one and onto mapping of the natural numbers onto the even natural...
Hi guys! I am starting to study Hubbard model with application in DFT and I have some doubts how to solve the Hubbard Hamiltonian. I have the DFT modeled to Hubbard, where the homogeneous Hamiltonian is
$$ H = -t\sum_{\langle i,j \rangle}\sigma (\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma} +...
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 ##...
(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)?
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...
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...
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}}\\...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
\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...
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...
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...
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...
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...
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...
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...
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...
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
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...
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...
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...
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...
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 &...
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...