Let A be an n x n symmetric matrix. Let {f₁,...,fn} be an orthonormal basis of eigenvectors of A satisfyingAfj=Xjfj, for j = 1,..., n.(a) Show that A =12=ΣALET.j=1(This formula is called the spectral decomposition of A.)(Hint: compare the (i, j)-entry of Aff with the (i, j)-entry of A = PDPT.)j=1(b) Given polynomial p(x) = akr +ak-12-¹+...+ a₁x + ao, define the matrixp(A) = akA" +ak-ak-1Ajk-1 +...+ a₁A+aoIn.Show that p(X1),...,p(An) are the eigenvalues of p(A) and22p(A) = Σp(x)fff.j=1This question implies CA (A) = 0 if сл(x) (x - ₁)(x − λ₂) (An). Your proof leads to theCayley-Hamilton Theorem for symmetric matrices.Cayley-Hamilton Theorem:Let CA(z) be the characteristic polynomial of a square matrix A. Then CA(A) is the zero matrix.

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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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For each matrix A, calculate an orthonormal basis of eigenvectors and use it to diagonalize A. [o 1 (ii) A = |1 1 41 (i) A = -121 -12 1
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 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
2. Prove Cayley-Hamilton Theorem for a diagonalizable matrix, A: Let A be diagonalizable and p(x) be its characteristic polynomial; i.e. value of A. p(A) : = 0 for A, eigen- Show that p(A) = 0.
Find the coordinate matrix of x relative to the orthonormal basis B in R. 4 x = (15, 5, 20), B = {(0).(-3,0), (0, 0, 1)} [x] B =
Let A be the matrix of the linear transformation T:R2→R2 that reflects each point across the x1-axis. What are the eigenvectors and corresponding eigenvalues of A?
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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