LetA =(a) By calculating and factorizing C₁(2), the characteristic polynomial ofA, 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 directcalculation that P-¹AP = B where(c) Using the fact thatP = [u v] and B =BnPA0= (-1)" [ 7]for every integer n, find An explicitly in terms of n. Hint: Begin byusing the equality A = PBP-¹ to express A" in terms of P and B.

Answered: Image /qna-images/answer/f41a299c-3003-4c29-828c-ab6ab47917bb.jpg

Answered: Let A = 2 (a) By calculating and…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Similar questions

Suppose the matrix A is used to transform points in the plane iteratively. That is, given a point v, consider the sequence vn = Anv. Letting U = [u1 u2] so that ui is an eigenvector associated to λi and letting v = c1u1 + c2u2 what is a simple expressions for an and bn so that vn = Anv = anu1 + bnu2.
Consider the vectors in C2 (sometimes called spinors) :(cos(0/2)e-ie/2 sin(8/2)et*/2 sin(8/2)e-i$/2 ) . V2 - cos(8/2)e*#/2 (i) First show that they are normalized and orthonormal. (ii) Assume that for the vector vị is an eigenvector with the corresponding eigenvalue +1 and that the vector v2 is an eigenvector with the corresponding eigenvalue -1. Apply the spectral theorem to find the corresponding 2 x 2 matrix. (iii) Since the vectors vị and v2 form an orthonormal basis in C? we can form an orthonormal basis in C4 via the Kronecker product W11 = V1 ® V1, W12 = V1 ® v2, W21 = V2 ® V1, W22 = V2 ® V2. Assume that the eigenvalue for w11 is +1, for w12 –1, for w21 -1 and for w22 +1. Apply the spectral theorem to find the corresponding 4 x 4 matrix.
Just part (c) please!
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
Algebra Question
(3) Let matrix A have eigenvectors . . associated with eigenvalues Xo = 4, A₁ = 2, A₂ = 0), respectively. Suppose some vector 7 is: 7 = №n +27₁ +30¹₂ Compute Ar. Give your result as a linear combination of eigenvectors ₁.
a) Confirm by multiplication that x is an eigenvector of A, and find the eigenvalue, [5 A = :X = b) Determine the following set of vectors is orthonormal. If a set is only orthogonal, normalize the vectors to produce an orthonormal set,
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.
7. Let A be an n x n Hermitian matrix. (a) Show that (Au, v) = (u, Av) for any n-dim vectors u and v. Here (, ·) denotes the inner product. (Hint: rewrite the inner product as matrix multiplication.) (b) Let x be an eigenvector associated to the eigenvalue A. Based on part (a), show that A(x, x) = X(x, x). (c) As in part (b), show that = X; that is, the eigenvalue A is real. 1
Let M be a 2 x 2 matrix with eigenvalues A₁ = 0.3, A2 = -0.25 with corresponding eigenvectors = [2²] ²2² = [9] · V2 Consider the difference equation with initial condition xo That is, write xo = C1V1 + C₂V2 H Write the initial condition as a linear combination of the eigenvectors of M. In general, X = Specifically, X4 V1 = For large k, xk Xk+1 = V₁+ )k v₁+ Mxk A V2 7)k V₂
Q2

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