is an eigen-ket of the Hamiltonian and E is its eigen-value, the bra-vector of the5. Ifquantum object can be written asEv) =exp-i-A.Ev) =cxp|В.2Ev) =exp|C.expD.E. None of the aboveNote: r is the space coordinate and t is the time.

Here |A> is an Eigen-ket of the Hamiltonian and E is its eigenvalue.

Answered: 5. If l*/ is an eigen-ket of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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5. Let S be a summetric matrix with eigenvalues A,...A. (counted with multiplicity). Order the eigenvalues so that |2 geglA,| > 0 = A,+1 == A.. a) Show that the singular values of S are Al,.A,. In particular, rank(S) = r. b) Suppose that S = QDQ", where Q is orthogonal and D is the diagonal matrix with diagonal entries A,.., A. Show that S has a singular eigenvalue decomposition of the form UEQ" (ie., V = Q). How is E related to D? How is U related to Q? e) Show that S = QDQ" is a singular value decomposition if and only if S is positive semi-definite. %3D
n . If B' has m rat independent eigenneetor, then prove that B has independent eiyenveetor.
26. Prove: If {u1, u2, . . . , u,} is an orthonormal basis for R", and if A can be expressed as A = c,u,u + c,u,u + · · + c,,4,U, then A is symmetric and has eigenvalues c1, C2, . . . , Cn .
6. Consider the matrix A = Compute the eigenvalues of A, and the corresponding eigenspaces.
lo 2 . [3 Given tensor [R] 07 find the eigenvalues and the three = 10 2 corresponding eigenvectors.
5. Let T: R2 → R2 be defined by T = and matrix A -x y ( []) = [- ² + 1] - [TB where B is the standard basis for R2 = i) Find the eigenvalues of A ii) Find the corresponding eigenvectors F [2 2 [² √3 = [21] = V. B
10. Suppose that |a) = E"-1 ax|k), |B) = E"-, bu|k). Show that (la)(8|)aj = a,b,, and that this means that the matrix representation in the given orthonormal basis satisfies |a)(3| = a b°, where a= [a a,]", b= [b, ..
4. Find the eigenvalues and a basis for each eigenspace in C². Show all of the work (no calculator). -2 [13] 2
6. Let A be an n x n hermitian matrix, i.e. A = A*. Assume that all n eigenvalues are different. Then the normalized eigenvectors { v; : j = 1,...,n} form an orthonormal basis in C". Consider B:= (Ax – ux, Ax – vx) = (Ax – ux)*(Ax – vx) | where (, ) denotes the scalar product in C" and µ,v are real constants with µ 0.
7.1 Venify that di is an corruponding eigenvector. eigenvalue of A and that X; is a A = - 3, Xi = (1, 0) Aュ --3, X,=() Axi= F]- Aメ。 = AュX。 0 -3
3. For nx n matrix A. define R₁ = one of the Gerschgorin's disks {z: True False "=1,i+j lajj 1, then each eigenvalue of A is either in aij | ≤ Ri}. z-
In each case, determine whether the vector v is an eigenvector of the 3 by 3 matrix A. No A = No A = No A = 9 -5 18 -8 6 -18 -4 5 -13 -7 -2 3 17 2 -13 -12 -2 8 -1 -4 -8 10 3 -2 -5 -4 -4 1. V = 2. " 3. υ: || 3 7 1 -1 1 -1 7 B 4 1

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5. If l*/ is an eigen-ket of the Hamiltonian and E is its eigen-value, the bra-vector of the quantum object can be written as E ) =exp| A. E еxp В. 2E v) =exp| С. D. E. None of the above Note: r is the space coordinate and t is the time.