True or False? Circle your answer and provide a justification for your choice.(a) T F: Any non-zero vector u e R" is an eigenvector of A = uu".(b) T F: For any n x n permutation matrix P, x = (1, 1,..., 1) E R" is an eigenvector of P.(c) T F: If 0 is an eigenvalue of an n x n matrix A, then N(A) = {0}.(d) T F: If A is an eigenvalue of a square matrix A, then every non-zero element in N(A – AI)is an eigenvector of A with eigenvalue A.(e) T F: If A is an eigenvalue of an n x n matrix A and u is an eigenvalue of an n x n matrixB, then Au is an eigenvalue of AB.

Let prove the result for n = 2 Let u = ab Then A=uuT = ababA=uuT =a2ababb2 u is the eigenvector of…

(a) T F: Any non-zero vector u e R" is an eigenvector of A = uu". (b) T F: For any n x n permutation matrix P, x = (1,1,..., 1) € R" is an eigenvector of P. (c) T F: If 0 is an eigenvalue of an n x n matrix A, then N(A) = {0}.

(a) T F: Any non-zero vector u e R" is an eigenvector of A = uu". (b) T F: For any n x n permutation matrix P, x = (1,1,..., 1) € R" is an eigenvector of P. (c) T F: If 0 is an eigenvalue of an n x n matrix A, then N(A) = {0}.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Similar questions

Suppose x E R" is an eigenvector of matrix A E R"X" associated with eigenvalue 1. Also suppose that the same vector x is an eigenvector of matrix B E RXn associated with eigenvalue y. Is x always an eigenvector of the matrix product AB? If so, derive the corresponding eigenvalue.
3. Let A be a nxn. matrix with distinct and positive eigenvalues. For each i, 1 ≤ i ≤n, let v; be an eigenvector of A with eigenvalue A, such that the v, are mutually orthogonal unit vectors. That is, (a) Suppose that w = j = 1,..., n. 1, for i = j, 0, for i j. av, for some a, E R. Prove that wv; = a; for all Vi Vj = (b) Show that x- (Ax) ≥ 0 for all x ER".
q7
(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.
Let the eigenvalues of the matrix A be A1, A2, . . A,n and the linearly w m w w w w w ma ndependent eigenvectors corresponding to these eigenvalues V1,V2,.. Vn. .... ww m ww (S [V1V2 . .. Vn]nxn) || by placing the vectors V1,V2,., Vn in each column of A is the eigenvalue of V1,V2,.., Vn placed on the w w m w m w w the S matrix; If the matrix diagonal, prove that, diagonal matrix. Also show that eAt mw wm m m w ww diag[A1 A2 . .. A,]nxn) with in the A = SAS-1 (A SeAt s-1 . %3D ww w w w w m
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Find the eigenvalues and corresponding eigenvector spaces for the following matrices
Mark as TRUE or FALSE. You will need to give an explanation. (a) If v1, v2 ..., Vn are linearly independent, then v1, V2 are linearly independent. (b) If v and w are two eigenvectors for the same eigenvalue c, then the sum v + w is also an eigenvector for the eigenvalue c. (c) Any 2 x 2 matrix has a real eigenvalue. (d) If v is an eigenvector for a matrix A and a matrix B then v is an eigenvector for A+ B.
For any n x n matrix A, x = 0 is an eigenvector. O True O False
9. Determine which of the following vectors 1 -1 is an eigenvector of the matrix 1 1 1 1 -1 1 -1 W = -1 -1 1 -1 -1 1 and find the corresponding eigenvalue.