Answered: Consider the n x n symmetric matrix H =…

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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Consider the n x n symmetric matrix H = I - 2uu^T, where u is a unit vector in Rⁿ, i.e., u^Tu = 1. H is an orthogonal matrix, and u is an eigenvector of H with an eigenvalue of -1.

(a) Let v be any nonzero vector perpendicular to u. Show that v is an eigenvector of H and find the corresponding eigenvalue. How many times is that eigenvalue repeated?

(b) Find the sum of the diagonal entries of H, find the sum of the eigenvalues of H.

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Consider the n x n symmetric matrix $H = I - 2 {uu}^{T}$ where u is a unit vector in Rⁿ, $(i . e) u^{T} u = 1$ . H is an orthogonal matrix, and u is an eigenvector of H with an eigenvalue of -1.

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