Suppose dhal we'dh elemen-ts a, , dg, -- A= SAS, ahex ^ cs a diagonal diagonal Asi = diSi , i = 1,2, A=SAS-! matrcx An. ノ (a) show thet ---- -, n (b) Show that ct x= d,S, + ą Sz t + cdy Sn Ohen --- 2 A x = |入く1 bor ie l, n. as k→ ∞.' Explain. that --- (C) Suppose what happens to A"x
Suppose dhal we'dh elemen-ts a, , dg, -- A= SAS, ahex ^ cs a diagonal diagonal Asi = diSi , i = 1,2, A=SAS-! matrcx An. ノ (a) show thet ---- -, n (b) Show that ct x= d,S, + ą Sz t + cdy Sn Ohen --- 2 A x = |入く1 bor ie l, n. as k→ ∞.' Explain. that --- (C) Suppose what happens to A"x
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.4: The Singular Value Decomposition
Problem 25EQ
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![**Topic: Matrix Diagonalization and Powers**
**Question:**
Suppose that \( A = S \Lambda S^{-1} \), where \( \Lambda \) is a diagonal matrix with diagonal elements \( \lambda_1, \lambda_2, \ldots, \lambda_n \).
(a) Show that \( A s_i = \lambda_i s_i; \, i = 1, 2, \ldots, n \).
(b) Show that if \( x = \alpha_1 s_1 + \alpha_2 s_2 + \ldots + \alpha_n s_n \), then
\[ A^k x = \alpha_1 \lambda_1^k s_1 + \alpha_2 \lambda_2^k s_2 + \ldots + \alpha_n \lambda_n^k s_n. \]
(c) Suppose that \( |\lambda_i| < 1 \) for \( i = 1, \ldots, n \). What happens to \( A^k x \) as \( k \to \infty \)? Explain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc40ed8c4-e2aa-4488-bef1-34c2accebf46%2F77182cc2-ed2e-4ec4-a8d9-90aa4c1c2383%2Fr6qxgg_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Matrix Diagonalization and Powers**
**Question:**
Suppose that \( A = S \Lambda S^{-1} \), where \( \Lambda \) is a diagonal matrix with diagonal elements \( \lambda_1, \lambda_2, \ldots, \lambda_n \).
(a) Show that \( A s_i = \lambda_i s_i; \, i = 1, 2, \ldots, n \).
(b) Show that if \( x = \alpha_1 s_1 + \alpha_2 s_2 + \ldots + \alpha_n s_n \), then
\[ A^k x = \alpha_1 \lambda_1^k s_1 + \alpha_2 \lambda_2^k s_2 + \ldots + \alpha_n \lambda_n^k s_n. \]
(c) Suppose that \( |\lambda_i| < 1 \) for \( i = 1, \ldots, n \). What happens to \( A^k x \) as \( k \to \infty \)? Explain.
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