7. For R, let A = Ag = cos sin sin 0 cos sin 0 and B Be= 86). cos cos sin (a) Show that B has an eigenvector v = (cos(0/2), sin(0/2)) with eigenvalue 1, and a second, orthogonal, eigenvector with eigen- value -1. Conclude that this matrix represents a reflection in the line tha li 9/9

For the system of differential equations, = A₁, A₂ = U₁ = U2 = y' a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) = [16 b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. = -16 21 C1 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 X₂ . Enter the eigenvectors as a matrix with an appropriate size. 14 19 y d) Determine a general solution to the system. Enter your answer in the format y(t) = c₁f₁(t)ví + c₂f₂(t)v₂ . y(t) = + C₂

7.'., Given A = Find the eigenvalues and eigenvectors of matrix A

2. Show that the wavefunction (eigenfunction) of a state a) can be expressed in position basis as (x/a).

I try to use the eigenvalue to find eigenvector, and I've already found the relation between x and y. I can substitute x=1 and get y=-i, but the solution provided is substituting x=-1 and get  y=i. There is a problem, how can I know which value of x should I choose?

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)) . +

Suppose the 3 × 3 matrix A has eigenvalues λ1 = −6, λ2 = −2, and λ3 = 5.

Find all the eigenvalues and eigenfunctions of the Sturm-Liouville problem. Use the integral inner product to verify the orthogonality of the eigenfunctions. y" + y = 0, y (0) = 0, y' (4) = 0

The matrix has two distinct, real eigenvalues. a) b) The smaller eigenvalue is λ = The larger eigenvalue is - 4 A = (1₂¹) 4 and the corresponding eigenvector is a = and the corresponding eigenvector is * = (↓). (₁). with y = with y

Given that the matrix A has eigenvaluesX₁ = -5 with corresponding eigenvector ₁ = and A₂ = 3 with corresponding eigenvector 2 = A = 1 [:] , find A. [:]