(a) Let A = 5 3 -3 1 1] 3 1 -1 1 " which has characteristic polynomial C₁(x) = -(x - 2)²(x - 5). (i) From the polynomial C₁ (2) alone, what can one say about the geometric multiplicities of the eigenvalues 2 and 5? Be as specific as you can. (ii) Now find a basis for the eigenspace associated to the eigenvalue 2, and hence give the geometric multiplicity of that eigenvalue.

Which of the following are eigen functions of the operator d2/dx2 , find the corresponding eigen value? (i) Sin x  (ii) Sin2x      (iii) e2x

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

7. Find the eigenvalues and eigenspaces for (a) Each of the Pauli operators X, Y, Z; [cos 0 (b) the operator Se = sin 0 - co sin 0 - cos 0 (c) 0) (1].

(8) Show that the constant term of the characteristic polynomial |21 – A| is ±JA|. Using this fact, consider the case here A = 0 is an eigenvalue of A. What can you conclude about the invertibility of A, and why?

1. -2 -2 Let A = -2 1 -2 -2 -2 1. a) Determine the characteristic polynomial of A: 2 2 XA(1) = det(rl3 – A) = det 2 I - 1 2 2 2 I - 1 b) multiplicity. Factor XA(r) and determine the eigenvalues and their algebraic c) For each eigenvalue a find a basis of the eigenspace, E.(A). d) Explain why the matrix is diagonaizable. e) P such that P-1AP = D. Write down a diagonal form, D, of A and a diagonalizing matrix

2: Let 10 -1 -1 A = 0. 10 10 -1 -1 10 (a) Find the characteristic polynomial of A. (b) Find the dimension of the eigenspace corresponding to the largest eigenvalue.

a.) The function, f(x) = 3X2-1, is an eigenfunction of the operator, (see image attched). Find the eigenvalue corresponding to this eigenfunction. b.) Show that the function is an eigenfunction of the operator, (See image attached) and calculate the eigenvalue.

Show v2  = [ 1  0 ]T is a corresponding “generalized eigenvector” of A

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