Unit 8 of 13
Linear Systems and Normal Modes
- 7 lessons
- Videos 42 to 48
Several coupled quantities form a system of first-order equations. This unit writes higher-order equations as systems and solves linear systems with eigenvalues, eigenvectors and matrix exponentials.
A state vector collects the unknowns, and a higher-order equation becomes a first-order system. For a constant matrix, real eigenvalues with an eigenbasis give modal solutions, complex eigenvalues give real oscillating solutions, and the eigenvalues sort planar phase portraits into nodes, saddles, spirals and centers. A repeated eigenvalue without enough eigenvectors needs a Jordan chain. Fundamental matrices and the matrix exponential carry every initial state forward, and variation of constants solves forced and time-dependent systems. Coupled oscillators separate into normal modes.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Systems and Eigenvalue Methods
Systems and Conversion from Higher-Order Equations
How a state vector collects an unknown and its derivatives, how a higher-order equation becomes a first-order system with vector initial data, how linear systems take matrix form, and what makes a system autonomous.
Video 42
Real Eigenvalues and Linear-System Solutions
How eigenvalues and eigenvectors of a constant coefficient matrix give modal solutions, how an eigenbasis builds every solution of a linear system, and how a starting state fixes the modal coefficients.
Video 43
Complex Eigenvalues and Planar Phase Portraits
How a complex eigenvector of a real matrix yields two real solutions, and how the eigenvalues sort planar linear phase portraits into nodes, saddles, spirals and centers, with their stability and direction of turning.
Video 44
Repeated Eigenvalues and Generalized Eigenvectors
How algebraic and geometric multiplicity tell a diagonalizable matrix from a defective one, and how generalized eigenvectors, Jordan chains and the nilpotent part supply the missing solutions of a linear system.
Video 45
Matrix Exponentials and Forced Systems
Matrix Exponentials and Fundamental Matrices
How the matrix exponential and fundamental matrices carry every initial state of a linear system forward, what Liouville's formula says about their determinants, and why noncommuting coefficient matrices need time order.
Video 46
Nonhomogeneous and Time-Dependent Linear Systems
How variation of constants and the inverse fundamental matrix solve a nonhomogeneous linear system, why a time-dependent coefficient matrix needs its own fundamental matrix, and why non-elementary integrals give exact answers.
Video 47
Coupled Oscillators and Normal Modes
Coupled Oscillators and Normal Modes
How springs that join several masses lead to a mass matrix and a stiffness matrix, how the generalized eigenvalue problem gives normal modes, and how modal coordinates separate the motion.
Video 48