4. Let V = C3 and define 0 : V → V by •(: )-( a -15a + 24b – 48c — ва + 10b — 18c 2а — 36 + 7с Compute the characteristic polynomial of 0 and hence find its eigenvalues. For each eigenvalue A compute the eigenspace Ex and write down a basis of it. Hence decide whether there exists a basis of V consisting of eigenvectors for 0.

4. Let V = C3 and define 0 : V → V by •(: )-( a -15a + 24b – 48c — ва + 10b — 18c 2а — 36 + 7с Compute the characteristic polynomial of 0 and hence find its eigenvalues. For each eigenvalue A compute the eigenspace Ex and write down a basis of it. Hence decide whether there exists a basis of V consisting of eigenvectors for 0.

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

Let A = Find the Characteristic Polynomial, and compute the Eigenvalues: They are (note i = v-1) is the iaginary "unit"): O p(A) = X² + 21 +1=A1 = -1, and A2 = -1, O p(A) = X² – 1=\1 = -1, and A2 = 1 O p(A) = X² – A +1=A1 = 1, and dz O p(A) = X² + 2A + 2 = A1 = -1+ i, and X2 = -1 - i O p(A) = X – 2A + 2 A1 = 1 + i, and A, = 1 – i O p(A) = X2 +1=\1 = i, and A2 = -i 1 W 000 000 F4 F3 E5
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 .
Consider the multiplication operator LA : C2 → C2 where -5 A -5 Find a basis B for a one dimensional LA-İnvariant subspace. B= { }
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
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.
For the system of differential equations, X1, X2 a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. U₁ = U2 = = y' = c) Find the eigenvectors assuming u₁ is the eigenvector associated with the smaller eigenvalue X₁ and u2 is the eigenvector associated with the larger eigenvalue A2. 5 2 - 1 Y2 (t) d) Determine two linearly independent solutions to the system. Enter the first solution in the format y₁ (t) y = = y₁ (t) Enter the second solution in the format y₂(t) = ƒ2(t) (v3h₁(t) + v₁h₂(t)) = : fi(t) (vigi(t) – v2g2(t)) . +
please solve it as soon as possible
Let A = (a) By calculating and factorizing C₁(2), the characteristic polynomial of A, show that -1 is the only eigenvalue of A, and then find a corre- sponding eigenvector u. -9 2 (b) Find a vector v R2 such that (A + I)v = u, and show by direct calculation that P-¹AP = B where (c) Using the fact that P = [u v] and B = Bn PA 0 = (-1)" [ 7] for every integer n, find An explicitly in terms of n. Hint: Begin by using the equality A = PBP-¹ to express A" in terms of P and B.
1)Be (1 1 -1 A = 2 0 1 1 1 0 a)Calculate the eigenvalues of A. b)Find the minimal polynomial of A and say whether 4 is diagonalizable.