Complex Number Eigenvalues & Eigenvectors Generate Spiral Sinks & Spiral Sources, Euler's Identity

Differential Equations and Linear Algebra Course, Lecture 23A.

(a.k.a. Differential Equations with Linear Algebra, Lecture 23A, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 23A).

#differentialequations #complexeigenvalues #eulersidentity

(0:00) This is the heart of the course
(0:40) Spiral sink example
(2:24) The imaginary unit i (satisfies i^2 = -1)
(2:57) Is this “legal”?
(4:26) Adding complex numbers
(5:31) Multiplying complex numbers
(8:37) Eigenvalues are complex conjugates of each other
(10:57) Corresponding complex eigenvector for one of the complex eigenvalues
(16:08) Draw the phase portrait using nullclines
(19:33) The origin is a spiral sink
(20:39) The most beautiful equation in the universe
(22:34) Euler’s Identity
(23:12) Verify Euler's Identity using Taylor series
(26:56) Complex solution of the system of ODEs
(29:18) Rearrange to obtain two real linearly independent solutions Yre and Yim.
(32:11) General solution (based on the solution space being 2-dimensional and the Basis Theorem)
(34:31) Why does this work?
(38:26) Summary of behavior of two-dimensional linear systems based on eigenvalues
(40:11) The Trace of a square matrix
(41:04) If a real matrix has non-real complex eigenvalues, they will be complex conjugates of each other.

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[georgesadler]

Professor Kinney, thank you for explaining Complex Number Eigenvalues/Vectors that Generate Spiral Sinks/Sources. This is the first time that I can remember seeing the computation of Eigenvectors with complex numbers. Euler's Identity is well known in all levels of Mathematics, Science and Engineering. I understand this video/lecture from start to finish.