1(a)Define T :12→2 by Tx(n)=-x(n), n=1,2,...Show that T is compact and positive with1eigenvaluesn=1, 2, ...n

Given: T :l2→l2 defined by Txn = 1nxn , n=1,2,…. We have to show that T is compact and positive with…

Answered: (a) Define T: R→P by Tx(n)= x(n),…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

The eigenvalues of the initial value problem y' +Ây=0, y(0)=0, y'(7)=0 are a) , = vn, for n=1,2,3,... b) 1, =n², for n=1,2,3,... c) 2, = 2n, for n=1,2,3,... d) , =, for n=1,2,3,... 2'
6. Find a basis of the null space of each of the differential operators below, and verify that the dimension of the null space is equal to the degree of the operator. (Use complex numbers where necessary for simplicity.) (a) D² − 4D + 3 (b) D² + 2D + 2 (c) D³ (d) Dn (e) Dª + 4.
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
4. Let At be a matrix that depends on r and is given by At t 3 -2 1 (a) For which te R matrix At has no real eigenvalues for At? (b) For which t E R matrix At has exactly one real distinct eigenvalue? Give that eigenvalue. (c) For which te R matrix At has two different real eigenvalues? Give one such eigenvalue as a function of t.
12- The system x = x + 4y, y' = y + x has two eigenvectors....
What are the eigenvalues and eigenfunctions: x′′ + λx = 0, x(1) = x(3), x′(1) = x′(3) - Definey by x(z) = y((z−2)π) for 1 ≤ z ≤ 3. Show that y(−π) = y(π) and y′(−π) = y′(π) - Substitute into the equation to get y′′((z−2)π) + (λ/π2) y((z−2)π), for 1 ≤ z ≤ 3 - Use the change of variable t = (z − 2)π to show that the above equation has a non-zero solution if and only if either λ = k2π2 for some integer k ≥ 1 or λ = 0 and the solutions (eigenfunctions) are given by cos(kt), sin(kt) and 1 for −π ≤ t ≤ π. - Plug back t = (z − 2)π to find the eigenfunctions and eigenvalues of original equation
i need the answer quickly
Let n ≥ 2. Consider the reversing operator R: R" → Rn defined by R(a₁, 02, ..., an-1, an) = (an, an-1,..., a2, 0₁). We denote by E₁ and E-1 the eigenspaces of eigenvalues X = 1 and λ = -1, respec- tively. (1) (2) If n is even, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable. If n is odd, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable.
Q7. If u = (1 + i, 1 – i,1+ 2i) and v = (i, 1, i), calculate ||u||, ||v||, and u. v Ans: ||u|| = 3, |v|| = V3, u· v = u#v = 4 – 3i Q8. Find the eigenvalues and eigenvectors of A = Ans: 11,2 = i,i. END
I need help with question and show your work need as soon as possbile
7. Let the 2 x 2 matrix A have real, distinct eigenvalues and μ. Suppose that an eigenvector of A is (1,0)" and an eigenvector of u is (-1,1)". Sketch the phase portraits of is x = Ax for the following cases: (1) 0 0. وچستان ستان ج
7. Let be the operator on P[r]3 defined by L(p(x)) = xp'(x) +p"(x) (1) Find the matrix A representing with respect to [1, x, x²]. (2) Find the matrix B representing with respect to [1, x, 1 + x²]. (3) Find the matrix S such that B S-¹ AS. (4) If p(x) = ao + a₁x + a₂(1 + x²), = calculate L¹ (p(x)).

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

(a) Define T: R→P by Tx(n)= x(n), n=1,2,.. Show that T is compact and positive with 1 eigenvalues , n=1, 2, .. n