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3. Let A be a n x n 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 = (b) Show that x (Ax) > 0 for all x ER".

3. Let A be a n x n 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 = (b) Show that x (Ax) > 0 for all x ER".

Advanced Engineering Mathematics
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 n x n matrix with distinct and positive eigenvalues. For each i, 1 ≤ i ≤n, let v; be an eigenvector of A with eigenvalue X, such that the v; are mutually orthogonal unit vectors. That is, ={ Vį. Vj = 0, for i = j, for i j. n (a) Suppose that w = Ei=1 Vi for some a; E R. Prove that wv; = a; for all j= 1,..., n. (b) Show that x (Ax) ≥ 0 for all x ER".
Transcribed Image Text:3. Let A be a n x n matrix with distinct and positive eigenvalues. For each i, 1 ≤ i ≤n, let v; be an eigenvector of A with eigenvalue X, such that the v; are mutually orthogonal unit vectors. That is, ={ Vį. Vj = 0, for i = j, for i j. n (a) Suppose that w = Ei=1 Vi for some a; E R. Prove that wv; = a; for all j= 1,..., n. (b) Show that x (Ax) ≥ 0 for all x ER".
(e) Find a basis for the image of T, Im(T), and state its dimension.
Transcribed Image Text:(e) Find a basis for the image of T, Im(T), and state its dimension.
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