Symmetric Matrix 2x2 Example: Orthogonal Diagonalization with Orthonormal Eigenvectors, Change Vars

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Bill Kinney's Differential Equations and Linear Algebra Course, Lecture 27B.

(a.k.a. Differential Equations with Linear Algebra, Lecture 27B, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 27B).

#linearalgebra #symmetricmatrices #diagonalization

(0:00) What to expect for the rest of the course
(1:58) Change of variables example (a symmetric matrix where the origin is a source for the corresponding linear system of ordinary differential equations)
(7:02) Orthonormal basis for the change of variables (change of coordinates matrix P). Make them unit vectors as well as orthogonal.
(12:19) Definition of an orthogonal matrix.
(13:38) Matrix exponential e^(t*A) can be computed using a similar product: e^(t*A) = Pe^(t*D)P^T
(16:43) Visualize the change of coordinates
(23:05) Mathematica
(25:38) Animation of rotation for the change of coordinates
(28:12) Another approach: make two pictures and show how the transformation maps a disk from one plane to the other

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Professor Kinney, thank you for an incredible Introduction to the Symmetric 2 by 2 Matrices, Orthogonal Diagonalization with Orthonormal Eigenvectors and the Change of Variables. These are topics/tools from Linear Algebra that is used to Solved Differential Equations. Matrix Exponential also plays a solid role in Linear Systems/Differential Equations.

georgesadler