The Characteristic Equation of a Square Matrix (11/18/05) To this point we know how to check whether a given number is an eigenvalue of a given square.

Slides:

Advertisements

Similar presentations

6.1 Eigenvalues and Diagonalization. Definitions A is n x n. is an eigenvalue of A if AX = X has non zero solutions X (called eigenvectors) If is an eigenvalue.

Advertisements

Section 9.2 Systems of Equations

Section 5.1 Eigenvectors and Eigenvalues. Eigenvectors and Eigenvalues Useful throughout pure and applied mathematics. Used to study difference equations.

The Inverse of a Matrix (10/14/05) If A is a square (say n by n) matrix and if there is an n by n matrix C such that C A = A C = I n, then C is called.

Eigenvalues and Eigenvectors (11/17/04) We know that every linear transformation T fixes the 0 vector (in R n, fixes the origin). But are there other subspaces.

Determinants (10/18/04) We learned previously the determinant of the 2 by 2 matrix A = is the number a d – b c. We need now to learn how to compute the.

Eigenvalues and Eigenvectors

Finding Zeros Given the Graph of a Polynomial Function Chapter 5.6.

5.II. Similarity 5.II.1. Definition and Examples

Chapter 2 Section 2 Solving a System of Linear Equations II.

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

Row Reduction and Echelon Forms (9/9/05) A matrix is in echelon form if: All nonzero rows are above any all-zero rows. Each leading entry of a row is in.

Orthogonal Sets (12/2/05) Recall that “orthogonal” matches the geometric idea of “perpendicular”. Definition. A set of vectors u 1,u 2,…,u p in R n is.

4.7 Inverse Matrices and Systems. 1) Inverse Matrices and Systems of Equations You have solved systems of equations using graphing, substitution, elimination…oh.

TODAY IN ALGEBRA…  Warm up: Find products of special polynomials  Learning Target: 9.4 You will solve polynomial equations in factored form  Independent.

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

Gerschgorin Circle Theorem. Eigenvalues In linear algebra Eigenvalues are defined for a square matrix M. An Eigenvalue for the matrix M is a scalar such.

Eigenvalues and Eigenvectors

Determinants In this chapter we will study “determinants” or, more precisely, “determinant functions.” Unlike real-valued functions, such as f(x)=x 2,

Domain Range definition: T is a linear transformation, EIGENVECTOR EIGENVALUE.

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.

Chapter 5 Eigenvalues and Eigenvectors 大葉大學資訊工程系黃鈴玲 Linear Algebra.

Triangular Form and Gaussian Elimination Boldly on to Sec. 7.3a… HW: p odd.

Pre-Calculus Section 1.5 Equations Objectives: To solve quadratics by factoring, completing the square, and using the quadratic formula. To use the discriminant.

Chapter 3 Determinants Linear Algebra. Ch03_2 3.1 Introduction to Determinants Definition The determinant of a 2  2 matrix A is denoted |A| and is given.

Scientific Computing General Least Squares. Polynomial Least Squares Polynomial Least Squares: We assume that the class of functions is the class of all.

Section 10.3 and Section 9.3 Systems of Equations and Inverses of Matrices.

Diagonalization and Similar Matrices In Section 4.2 we showed how to compute eigenpairs (,p) of a matrix A by determining the roots of the characteristic.

Solving Linear Systems Solving linear systems Ax = b is one part of numerical linear algebra, and involves manipulating the rows of a matrix. Solving linear.

Elementary Linear Algebra Anton & Rorres, 9 th Edition Lecture Set – 07 Chapter 7: Eigenvalues, Eigenvectors.

Eigenvalues The eigenvalue problem is to determine the nontrivial solutions of the equation Ax= x where A is an n-by-n matrix, x is a length n column.

Section 4Chapter 4. 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives Solving Systems of Linear Equations by Matrix Methods Define.

Section 6.4 Solving Polynomial Equations Obj: to solve polynomial equations.

4.7 Solving Systems using Matrix Equations and Inverses

5.1 Eigenvectors and Eigenvalues 5. Eigenvalues and Eigenvectors.

PreCalculus Section P.1 Solving Equations. Equations and Solutions of Equations An equation that is true for every real number in the domain of the variable.

2.5 Determinants and Multiplicative Inverses of Matrices. Objectives: 1.Evaluate determinants. 2.Find the inverses of matrices. 3.Solve systems of equations.

Section 1.7 Linear Independence and Nonsingular Matrices

Multiply Matrices Chapter 3.6. Matrix Multiplication Matrix multiplication is defined differently than matrix addition The matrices need not be of the.

Section 9.3 Systems of Linear Equations in Several Variables Objectives: Solving systems of linear equations using Gaussian elimination.

Solving One Step Equations Algebra I. Addition and Subtraction One Step Equations A solution of an equation is the value or values of the variable that.

Chapter 8 Systems of Linear Equations in Two Variables Section 8.3.

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

The Matrix Equation A x = b (9/16/05) Definition. If A is an m  n matrix with columns a 1, a 2,…, a n and x is a vector in R n, then the product of A.

Review of Eigenvectors and Eigenvalues from CliffsNotes Online mining-the-Eigenvectors-of-a- Matrix.topicArticleId-20807,articleId-

Characteristic Polynomial Hung-yi Lee. Outline Last lecture: Given eigenvalues, we know how to find eigenvectors or eigenspaces Check eigenvalues This.

Review of Eigenvectors and Eigenvalues

1. The square root of 9 is 3 True

Elementary Linear Algebra Anton & Rorres, 9th Edition

Section 4.1 Eigenvalues and Eigenvectors

Solving Systems Using Substitution

Euclidean Inner Product on Rn

Solving Equations by Factoring and Problem Solving

9.3 Solving Quadratic Equations

Eigenvalues and Eigenvectors

Class Notes 11.2 The Quadratic Formula.

Main Ideas Key Terms Chapter 2 Section 3 Graphing a Polynomial

Inverse Matrices and Systems

Eigenvalues and Eigenvectors

4.4 Objectives Day 1: Find the determinants of 2  2 and 3  3 matrices. Day 2: Use Cramer’s rule to solve systems of linear equations. Vocabulary Determinant:

EIGENVECTORS AND EIGENVALUES

Elementary Linear Algebra Anton & Rorres, 9th Edition

Section 8.2 Part 2 Solving Systems by Substitution

Inverse Matrices and Systems

6-7: Solving Radical Equations

Exercise Solve x – 14 = 35. x = 49.

Eigenvalues and Eigenvectors

Objective SWBAT solve polynomial equations in factored form.

Linear Algebra Lecture 28.

Equations With Variables on Both Sides

Presentation transcript:

The Characteristic Equation of a Square Matrix (11/18/05) To this point we know how to check whether a given number is an eigenvalue of a given square matrix A, but we do not know how to find the eigenvalues of A (unless A is upper triangular). The trick is to use the determinant of A !

Definition The characteristic equation of an n by n square matrix A is the equation det(A - I n ) = 0 This is a polynomial equation of degree n in the variable. Punch line: The roots of this equation are the eigenvalues of A.

Two examples Find the eigenvalues of A = Find the eigenvalues of B =

Why it works is an eigenvalue of a matrix A if and only if A x = x for some nonzero vector x, which is true if and only if (A - I ) x = 0 has a non-trivial solution, which is true if and only if det(A - I ) = 0. Is that cool or what?

But there are drawbacks… First, determinants are not easy to compute, as we well know. Second, once the determinant is found, the polynomial equation may not be easy to solve. For example, what are the eigenvalues of matrices whose characteristic equations are: = = 0

Assignment for Monday Finish up HW #3. Read Section 5.2. (Note: We already know the material on the determinant. They are assuming that the reader skipped Chapter 3) Do Exercises 1-11 odd, 15, and 19.