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Consider the (scalar) ODE '(t) = ax(t) with initial data (0) = IO.

Consider the (scalar) ODE '(t) = ax(t) with initial data (0) = IO.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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1. Consider the (scalar) ODE x'(t) = ax(t) with initial data (0) = o. Show that r(t) solves the preceding equation. Let B be an n x n matriz, with real or complex entries, and I the n x n identity matriz. The matrix exponential is the n x n matrix given by the (convergent) sum B² B3 eBI+B+ + +...= 2 6 Similarly, we can define an n x n matrix which is a function of t by B² 2 etBI+tB + t² B3 6 Bk Σk! k=0 +3 += 8 tk Bk k! Toeat Σ k=0
Transcribed Image Text:1. Consider the (scalar) ODE x'(t) = ax(t) with initial data (0) = o. Show that r(t) solves the preceding equation. Let B be an n x n matriz, with real or complex entries, and I the n x n identity matriz. The matrix exponential is the n x n matrix given by the (convergent) sum B² B3 eBI+B+ + +...= 2 6 Similarly, we can define an n x n matrix which is a function of t by B² 2 etBI+tB + t² B3 6 Bk Σk! k=0 +3 += 8 tk Bk k! Toeat Σ k=0
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