please explain all steps . 140 pints] Let A be a Hermitian operator with a complete...
-. Let Ä be an NỮN Hermitian operator corresponding to an observable in a quantum system whose Hilbert space is an N-dimensional one. Recall that the eigenvalues and eigenvectors of Ä are given by the solutions of Âlai) = ailai), i = 1, ..., N where the eigenvalues ai are all real, an the eigenvectors form a complete orthonormal set on the N-dimensional Hilbert space, meaning that (ailaj) = dij. Suppose the state vector of the system at some point...
2. The spectral decomposition theorem states that the eigenstates of any Hermitian matrix form an orthonormal basis for the linear space. Let us consider a real 3D space where a vector is denoted by a 3x1 column vector. Consider the symmetric matrix B-1 1 1 Show that the vectors 1,0, and1are eigenvectors of B, and find 0 their eigenvalues. Notice that these vectors are not orthogonal. (Of course they are not normalized but let's don't worry about it. You can...
Please help and explain all steps 9 Marks [5 0 0 1 8. Let A= 10 3 [0 0 -2] (a) Find all eigenvalues of A and their corresponding eigenvectors. (b) Is A diagonalizable? If so, find a matrix P and diagonal matrix D such that P-1AP = D.
Problem 4. Give an example of a linear operator T on a finite-dimensional vector space such that T is not nilpotent, but zero is the only eigenvalue of T. Characterize all such operators. Problem 5. Let A be an n × n matrix whose characteristic polynomial splits, γ be a cycle of generalized eigenvectors corresponding to an eigenvalue λ, and W be the subspace spanned by γ. Define γ′ to be the ordered set obtained from γ by reversing the...
Let V be a finite-dimensional inner product space. For an operator TEL(V), define its norm by ||T|:= max{||Tull VEV. ||0|| = 1}. (1) To explain this, note that {l|Tu ve V, || 0 || = 1} is a non-empty subset of [0,00). The expression max{||TV|| | V EV, ||0|| = 1} means the maximum, or largest, value in this set. In words, the norm of an operator describes the maximal amount that it 'stretches' (or shrinks) vectors. (a) (1 point)...
please explain thoroughly :) Determine whether each of the following sets is orthogonal, orthonormal, or neither A= 2- -J L2-1 Let U be an n × n matrix with orthonormal columns. Prove that det U-1.
please do (iv) and explain all the steps (4) Though I proved in class the orthogonality of eigenfunctions of the Sturm-Liouville BVP with respect to the weight function o when the Sturm-Liouville operator is regular, the orthogonality condition for eigenfunctions is true for many singular Sturm-Liouville BVP's. In this problem you will see an example. Consider then the singular Sturm-Liouville problem [(1 -u-u -1< r< 1, where u is required to be finite at ±1, meaning that limg+1 u(z) is...
Write a clear line, please explain the steps let R. is aring and R. is a ring 2 show that R. @ R is a ring ring? Z prove
Let h0= h1=1 and hn= 2hn-1+hn-2 for n >= 2. Prove that hn <= 2.5n. Please show the induction steps way. Show all steps Thanks!!
PLEASE SHOW ALL STEPS WITH EXPLAINATION Let m and n be positive integers and let k be the least common multiple of m and n. Show that mZ∩nZ=kZ.