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 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 →
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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