Complex Diagonalization, Real Canonical Form (Real Normal Form) for Matrices w/ Complex Eigenvalues скачать в хорошем качестве

Complex Diagonalization, Real Canonical Form (Real Normal Form) for Matrices w/ Complex Eigenvalues

Differential Equations, 4th Edition (by Blanchard, Devaney, and Hall): https://amzn.to/35Wxabr. Amazon Prime Student 6-Month Trial: https://amzn.to/3iUKwdP. At the start I look at the Diagonalization Theorem when there are n linearly independent real eigenvectors and why this works, based on the definition of matrix multiplication (compute the matrix products A*P and P*D to see that they are the same), the definition of eigenvalues and eigenvectors, and the Invertible Matrix Theorem. Then I do a complex diagonalization example (diagonalize a matrix with complex eigenvalues over the complex number field): given a real matrix with complex number eigenvalues, I find the complex eigenvectors and form a change of coordinates matrix P whose columns are the complex eigenvalues. The matrix P^(-1)*A*P is then a diagonal matrix D whose diagonal entries are the complex eigenvalues of A. Finally, and most importantly, I talk about the real canonical form (real normal form) for real matrices with complex eigenvalues. The complex eigenvectors are broken into their real and imaginary parts, and those real vectors form the columns of the change of coordinates matrix P. The matrix product P^(-1)*A*P is not a diagonal matrix, but it is in a "nice" canonical form (a "nice" normal form), which is the sum of a diagonal matrix and a skew symmetric matrix. Mathematica is used to visualize the change of coordinates (change of variables). Once we compute the matrix exponential of such a matrix, we can use it to solve a generic initial value problem dy/dt = A*Y, Y(0) = Y0, where A is the original matrix with the complex eigenvalues. Bill Kinney's Differential Equations and Linear Algebra Course, Lecture 28B. (a.k.a. Differential Equations with Linear Algebra, Lecture 28B, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 28B). #linearalgebra #complexnumbers #normalform Google drive link for Differential Equations and Linear Algebra course lecture documents: https://drive.google.com/drive/folder... Using Mathematica for ODEs (Ordinary Differential Equations) Playlist:    • Using Mathematica for Ordinary Differentia...   Visual Linear Algebra Online (at infinityisreallybig.com): https://infinityisreallybig.com/categ... Infinite Powers, How Calculus Reveals the Secrets of the Universe (by Steven Strogatz): https://amzn.to/2XXRCF6 Another Differential Equations lecture I made: Differential Equations: As Much As You Can Possibly Learn About in 50 Minutes, especially Population Models:    • Differential Equations Crash Course: As Mu...   Check out my blog: https://infinityisreallybig.com/ Bethel University is a Christian liberal arts university in St. Paul, Minnesota with strong science, engineering, mathematics and computer science departments. You can also get to know your professors personally. https://www.bethel.edu/ (0:00) Introduction (1:01) Diagonalization Theorem (2:26) Why is the Diagonalization Theorem true? (6:29) Complex eigenvalues and eigenvectors case (12:58) Real canonical form (13:44) Example: a real matrix with complex eigenvalues that is not in real canonical form) (19:16) Relationship to differential equations (22:50) The Phase Plane and Skew Coordinates (25:17) Mathematica calculations and animations AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.