Let A be an n x n symmetric matrix. Let {f₁,..., fn} be an orthonormal basis of eigenvectors of A satisfyingAf; = Ajfj, for j = 1,..., n.n(a) Show that A = - Σλ;fff.j=1compare the (i, j)-entry ofnj=1Aff with the (i, j)-entry of A = PDPT(b) Given polynomial p(x) = ax + ak-12k-1 + ax + ao, define the matrixp(A) = akAk +ak-1A-1 + a₁A+aoIn.Show that p(A₁),...,p(An) are the eigenvalues of p(A) andnp(A) = [p(\;)£;£³.j=1

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Answered: Let A be an n x n symmetric matrix. Let…

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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Let be an eigenvector of a matrix A corresponding to the eigenvalue 1, = -1 and the vector be an eigenvector of A corresponding to the eigenvalue 1, = 4. Which of the following is equal to A. e) L d-) e-) nndiğini görene kadar hekleviniz Soruvu bos birakmak isterseniz işarett
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".
Let M be a 2 × 2 matrix with eigenvalues ₁ = -0.8, 12 = 1 with corresponding eigenvectors Consider the difference equation with initial condition X = V₁ = V2 = Xk+1 = Mxk Write the initial condition as a linear combination of the eigenvectors of M. That is, write x0 = c1V1 + C2V2 = In general, X = Vi+ V2 k )* VI+ ) k V2 Specifically, X2 = For large k, xk →
Let M be a 2 x 2 matrix with eigenvalues A1 = -0.9, A2 =-0.75 with corresponding eigenvectors V1 V2 Consider the difference equation Mx: with initial condition xo Write the initial condition as a linear combination of the eigenvectors of M. That is, write xo = ¢¡V1 +C2V2 Vi+ V2 In general, X Vi+ )* v2 Specifically, x2 For large k, Xk →
5. Let S be a summetric matrix with eigenvalues A,...A. (counted with multiplicity). Order the eigenvalues so that |2 geglA,| > 0 = A,+1 == A.. a) Show that the singular values of S are Al,.A,. In particular, rank(S) = r. b) Suppose that S = QDQ", where Q is orthogonal and D is the diagonal matrix with diagonal entries A,.., A. Show that S has a singular eigenvalue decomposition of the form UEQ" (ie., V = Q). How is E related to D? How is U related to Q? e) Show that S = QDQ" is a singular value decomposition if and only if S is positive semi-definite. %3D
(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.
I need the next 3 parts pls!
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
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.
Find the eigenvalues and corresponding eigenvector spaces for the following matrices
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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