let A be a n×n matrix with distinct and positive eigenvalues. for each i, 1< i< n, let vi be an eigenvector of A with eigenvalue λi, such that the vi are mutually orthogonal unit vectors. that is vi. vj= { 1, for i=j, 0 for i≠j a) suppose that w=Σni=1 αivi for some αi∈R. prove that w.vj= αj for all j=1,..,n b) show that x. (Ax) > 0 for all x∈Rn

let A be a n×n matrix with distinct and positive eigenvalues. for each i, 1< i< n, let vi be an eigenvector of A with eigenvalue λi, such that the vi are mutually orthogonal unit vectors. that is vi. vj= { 1, for i=j, 0 for i≠j a) suppose that w=Σni=1 αivi for some αi∈R. prove that w.vj= αj for all j=1,..,n b) show that x. (Ax) > 0 for all x∈Rn

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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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".
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