3. For nx n matrix A. define R₁ =one of the Gerschgorin's disks {z:TrueFalse"=1,i+j lajj 1, then each eigenvalue of A is either inaij | ≤ Ri}.z-

Statement: For n×n matrix A. Define Ri=∑j=1,i≠jn|aij|, then, each eigenvalue of A is either in one…

Answered: 3. For n x n matrix A. define R₁ = one…

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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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".
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Let A be an invertible n x n matrix . Which of the following statements is not necessarily true Select one: a. 0 is an eigenvalue of A b. The equation Ax =b has at least one solution for each b in R" C. AT is invertible d. Column space of A = R"
Let A be a symmetric matrix with eigenvalues 3 and 0 and corresponding eigenvectors (a, b) and (c, d). a If the eigenvector matrix is S= then the eigenvectors of B = S-1 A S are O a. (1, 0), (0, 1) O b. (3, 3), (-3, -3) Oc. cannot be determined O d. (2, ), (-2. -i) O e. (2, 2), (-2, -2)
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.
46. Let A and B be n× n matrices, each with n distinct eigenvalues. Prove that A and B have the same eigenvectors if and only if AB = BA.
An ecologist is studying the impacts of air pollution on giant panda populations, specifically on panda birth rates. Due to the historical interest in giant pandas, the ecologist has access to data from the last 30 years that allows her to fit two models to the evolution of giant panda populations over time (one for 30 years and one for the last 5 years after a new chemical plant was built close to the panda habitat). She will use these models to determine the long-term impacts of this pollution on panda populations. She has the following information to help her model the two sets of population dynamics data on adult and baby pandas : i. 1) Without pollution, for every 1 adult female panda a year earlier, 0.75 baby pandas are born each year (reproduction rate = 0.75). 2) With pollution, for every 1 adult female panda a year earlier, 0.7 baby pandas are born each year (reproduction rate = 0.7). 3) 3 in 4 of the adult female pandas survive to the next year. 1 in 3 female baby pandas…
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.

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3. For n x n matrix A. define R₁ = one of the Gerschgorin's disks {z: Iz-a₁j | ≤ R}. O True O False "=1,i+j lajj 1, then each eigenvalue of A is either in