Homogeneous Linear Systems

Slides:

Advertisements

Similar presentations

Ch 7.6: Complex Eigenvalues

Advertisements

Ch 7.7: Fundamental Matrices

Chapter 6 Eigenvalues and Eigenvectors

Notes Over 4.5 Solving a Linear System Use an Inverse Matrix to solve the linear system.

Eigenvalues and Eigenvectors

Some useful linear algebra. Linearly independent vectors span(V): span of vector space V is all linear combinations of vectors v i, i.e.

Ch 7.8: Repeated Eigenvalues

5. Topic Method of Powers Stable Populations Linear Recurrences.

Ch 7.3: Systems of Linear Equations, Linear Independence, Eigenvalues

Ch 7.5: Homogeneous Linear Systems with Constant Coefficients

5 5.1 © 2012 Pearson Education, Inc. Eigenvalues and Eigenvectors EIGENVECTORS AND EIGENVALUES.

Boyce/DiPrima 9th ed, Ch 7.3: Systems of Linear Equations, Linear Independence, Eigenvalues Elementary Differential Equations and Boundary Value Problems,

7.2 Solving Recurrence Relations. Definition 1 (p. 460)- LHRR-K Def: A linear homogeneous recurrence relations of degree k with constant coefficients.

6.5 – Applying Systems of Linear Equations

5.4 – Solving Compound Inequalities. Ex. Solve and graph the solution.

Computing Eigen Information for Small Matrices The eigen equation can be rearranged as follows: Ax = x  Ax = I n x  Ax - I n x = 0  (A - I n )x = 0.

Section 5.1 First-Order Systems & Applications

Homogeneous Linear Systems with Constant Coefficients Solutions of Systems of ODEs.

Ch. 3.3: Graphing and Solving Systems of Linear Inequalities Goals Graph a system of linear inequalities to find the solutions of the system. Use systems.

Linear Algebra Diyako Ghaderyan 1 Contents:  Linear Equations in Linear Algebra  Matrix Algebra  Determinants  Vector Spaces  Eigenvalues.

Nonhomogeneous Linear Systems Undetermined Coefficients.

Linear Algebra Diyako Ghaderyan 1 Contents:  Linear Equations in Linear Algebra  Matrix Algebra  Determinants  Vector Spaces  Eigenvalues.

7.1 Eigenvalues and Eigenvectors

5.1 Eigenvectors and Eigenvalues 5. Eigenvalues and Eigenvectors.

5.1 Eigenvalues and Eigenvectors

5 5.1 © 2016 Pearson Education, Ltd. Eigenvalues and Eigenvectors EIGENVECTORS AND EIGENVALUES.

Differential Equations MTH 242 Lecture # 28 Dr. Manshoor Ahmed.

Tutorial 6. Eigenvalues & Eigenvectors Reminder: Eigenvectors A vector x invariant up to a scaling by λ to a multiplication by matrix A is called.

Chapter 6 Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

Systems of Differential Equations Phase Plane Analysis

Eigenvalues and Eigenvectors

ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department

Warm-up Problem Use the Laplace transform to solve the IVP.

Problem 1.5: For this problem, we need to figure out the length of the blue segment shown in the figure. This can be solved easily using similar triangles.

3.3 – Solving Systems of Inequalities by Graphing

Eigenvalues and Eigenvectors

The Inverse of a Square Matrix

Complex Eigenvalues kshum ENGG2420B.

Solutions to Systems Of Linear Equations II

Boyce/DiPrima 10th ed, Ch 7.7: Fundamental Matrices Elementary Differential Equations and Boundary Value Problems, 10th edition, by William E. Boyce and.

Systems of Ordinary Differential Equations Case I: real eigenvalues of multiplicity 1 MAT 275.

Eigenvalues and Eigenvectors

Some useful linear algebra

Systems of Differential Equations Nonhomogeneous Systems

Physics 321 Hour 29 Principal Axes.

Boyce/DiPrima 10th ed, Ch 7.3: Systems of Linear Equations, Linear Independence, Eigenvalues Elementary Differential Equations and Boundary Value Problems,

MATH 374 Lecture 23 Complex Eigenvalues.

MATH 374 Lecture 24 Repeated Eigenvalues.

Systems of Linear Equations

MAE 82 – Engineering Mathematics

Eigenvalues and Eigenvectors

Equivalent State Equations

Hour 33 Coupled Oscillators I

Linear Algebra Lecture 32.

Engineering Mathematics-I Eigenvalues and Eigenvectors

RAYAT SHIKSHAN SANSTHA’S S.M.JOSHI COLLEGE HADAPSAR, PUNE

Complex Eigenvalues and Non-Homogeneous Systems

Linear Algebra Lecture 30.

System of Linear First-Order DE

3.3 Notes – Graph Systems of Linear Inequalities

L3-3 Objective: Students will graph systems of linear inequalities

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

Principal Components What matters most?.

Chapter 2 Determinants.

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

Ch 7.5: Homogeneous Linear Systems with Constant Coefficients

Lin. indep eigenvectors One single eigenvector

Presentation transcript:

Homogeneous Linear Systems Thm. Let λ1, λ2, λ3 be distinct eigenvalues of A with corresponding eigenvectors K1, K2, K3. Then the general solution to X  = AX is

Ex. Solve

Ex. Solve

When eigenvalues are not distinct, things change If λ1 repeats twice and has two eigenvectors, the solution is

Ex. Solve

If λ repeats twice and has eigenvector K, the solutions are and where P is given by

Ex. Solve

If λ repeats three times and has eigenvector K, the solutions are , , where Q is given by