M334Lec32 - Math 334 Lecture #32 7.6: Complex Eigenvalues...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 334 Lecture #32 7.6: Complex Eigenvalues Finding Real Solutions. Suppose that a constant n n matrix A has a complex- conjugate pair of eigenvalues: r 1 = + i, r 2 = - i. If ~ (1) is a complex eigenvector corresponding to r 1 , then a complex eigenvector corre- sponding to r 2 = r 1 is ~ (2) = (1) (the component-wise complex conjugate): ( A- r 1 I ) ~ (1) = ~ ( A- r 1 I ) ~ (1) = ~ ( A- r 1 I ) ~ (1) = ~ ( A- r 2 I ) ~ (1) = ~ . Thus, two complex vector solutions of ~x = A~x are ~x (1) = ~ (1) e r 1 t , ~x (2) = ~ (1) e r 1 t = ~x (1) . [NOTE: once we know ~x (1) , we know ~x (2) !] By writing ~ (1) = ~a + i ~ b (where ~a and ~ b are real vectors), it follows from Eulers Formula that ~x (1) ( t ) = ~ (1) e r 1 t = ( ~a + i ~ b ) exp { ( + i ) t } = ( ~a + i ~ b ) e t e it = e t ( ~a + i ~ b )( cos t + i sin t ) = e t ~a cos t + i~a sin t + i ~ b cos t- ~ b sin t = e t ~a cos t- ~ b sin...
View Full Document

Page1 / 3

M334Lec32 - Math 334 Lecture #32 7.6: Complex Eigenvalues...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online