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 orthogonalunit 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 allVi Vj =(b) Show that x- (Ax) ≥ 0 for all x ER".

Given that A be a n×n matrix with distinct and positive eigenvalues. For each i, 1≤i≤n, let vi be an…

Answered: 3. Let A be a n x n matrix with…

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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Suppose u1, ... , Un and v1, ... , Vn are orthonormal bases for Rn. Construct the matrixA=UI:VT that transforms each Vj intouj to giveAv1 =u1, ... , Avn =Un.
2. Let A be an n x n nonsingular matrix with entries from a field F. Suppose that the vectors v1,... , V in F" are linearly independent. Show that AVg are linearly independent. the vectors AV1, .
(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.
5. Prove that the matrices A and A' have the same eigen values. [K Prove th prooteriotic root of a matrix iff the matrix
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
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"
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.
3. For nx n matrix A. define R₁ = one of the Gerschgorin's disks {z: True False "=1,i+j lajj 1, then each eigenvalue of A is either in aij | ≤ Ri}. z-
Write a formal proof for the given statements below. (b) Let A be an n × n matrix. Suppose that vector v ∈ Rn is an eigenvector of A with the corresponding eigenvalue λ ∈ R. Prove that vector v is an eigenvector of A3 with the corresponding eigenvalue λ3 .
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

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