Diagonalization Definition and 133 Threads

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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

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...

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...

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...

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=...

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

[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...

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...

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.

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

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)

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...

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...

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...

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

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:

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...

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

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...

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 |...

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

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...

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...