Theorem: If A has a repeated real root λ with corresponding eigenvector K, then all solutions of X' = A X are of the form: ( * ) - Ge" K₁+ = e¹¹ K₁+ c₂ ( t e¹¹ K₁ + ett (3) W1 where (A - I) = K₁. W2 Theorem: Let ₁ = α + i ẞ be a complex eigenvalue of the coefficient matrix A with corresponding eigenvector K₁ = B₁ + i B2. Then all solutions of X' = A X are of the form: x c₁ (B₁ cos ẞt - B₂ sin ẞt) eat + c2 (B2 cos ẞt + B₁ sin ẞt) eat. ν x That is: C₁ (Re(K₁) cos ẞt - Im(K₁) sin ẞt) eat + 2 (Im(K₁) cos ẞt + Re(K₁) sin ẞt) eat. y

Theorem: If A has a repeated real root λ with corresponding eigenvector K, then all solutions of X' = A X are of the form: ( * ) - Ge" K₁+ = e¹¹ K₁+ c₂ ( t e¹¹ K₁ + ett (3) W1 where (A - I) = K₁. W2 Theorem: Let ₁ = α + i ẞ be a complex eigenvalue of the coefficient matrix A with corresponding eigenvector K₁ = B₁ + i B2. Then all solutions of X' = A X are of the form: x c₁ (B₁ cos ẞt - B₂ sin ẞt) eat + c2 (B2 cos ẞt + B₁ sin ẞt) eat. ν x That is: C₁ (Re(K₁) cos ẞt - Im(K₁) sin ẞt) eat + 2 (Im(K₁) cos ẞt + Re(K₁) sin ẞt) eat. y

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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