NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN1 Lecture 28 is positive definite Similar matrices Jordan form.

Slides:

Advertisements

Similar presentations

Scientific Computing QR Factorization Part 2 – Algorithm to Find Eigenvalues.

Advertisements

Matrix Theory Background

OCE301 Part II: Linear Algebra lecture 4. Eigenvalue Problem Ax = y Ax = x occur frequently in engineering analysis (eigenvalue problem) Ax =  x [ A.

Refresher: Vector and Matrix Algebra Mike Kirkpatrick Department of Chemical Engineering FAMU-FSU College of Engineering.

Eigenvalues and Eigenvectors

Lecture 19 Singular Value Decomposition

Linear Transformations

ENGG2013 Unit 19 The principal axes theorem

5. Topic Method of Powers Stable Populations Linear Recurrences.

Matrix Operations. Matrix Notation Example Equality of Matrices.

Linear Algebra---Meiling CHEN

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN1 Lecture 15 Projection Least squares Projection matrix.

Lecture 18 Eigenvalue Problems II Shang-Hua Teng.

6 1 Linear Transformations. 6 2 Hopfield Network Questions.

Digital Control Systems Vector-Matrix Analysis. Definitions.

NUU Department of Electrical Engineering Linear Algedra ---Meiling CHEN1 Eigenvalues & eigenvectors Distinct eigenvalues Repeated eigenvalues –Independent.

Matrices CS485/685 Computer Vision Dr. George Bebis.

Table of Contents Matrices - Inverse Matrix Definition The inverse of matrix A is a matrix A -1 such that... and Note that... 1) For A to have an inverse,

5.1 Orthogonality.

1 Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 5QF Introduction to Vector and Matrix Operations Needed for the.

Introduction The central problems of Linear Algebra are to study the properties of matrices and to investigate the solutions of systems of linear equations.

Computer Vision Group Prof. Daniel Cremers Autonomous Navigation for Flying Robots Lecture 2.1: Recap on Linear Algebra Daniel Cremers Technische Universität.

Today’s class Boundary Value Problems Eigenvalue Problems

Linear Algebra Review 1 CS479/679 Pattern Recognition Dr. George Bebis.

Fundamentals from Linear Algebra Ghan S. Bhatt and Ali Sekmen Mathematical Sciences and Computer Science College of Engineering Tennessee State University.

8.1 Vector spaces A set of vector is said to form a linear vector space V Chapter 8 Matrices and vector spaces.

Eigenvalues and Eigenvectors

Therorem 1: Under what conditions a given matrix is diagonalizable ??? Jordan Block REMARK: Not all nxn matrices are diagonalizable A similar to (close.

Chapter 2 Systems of Linear Equations and Matrices

Inverse and Identity Matrices Can only be used for square matrices. (2x2, 3x3, etc.)

Class 27: Question 1 TRUE or FALSE: If P is a projection matrix of the form P=A(A T A) -1 A T then P is a symmetric matrix. 1. TRUE 2. FALSE.

Linear algebra: matrix Eigen-value Problems Eng. Hassan S. Migdadi Part 1.

1 Matrix Algebra and Random Vectors Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking.

Lecture 1: Math Review EEE2108: Optimization 서강대학교 전자공학과 2012 학년도 1 학기.

KEY THEOREMS KEY IDEASKEY ALGORITHMS LINKED TO EXAMPLES next.

5.1 Eigenvectors and Eigenvalues 5. Eigenvalues and Eigenvectors.

Stats & Summary. The Woodbury Theorem where the inverses.

Similar diagonalization of real symmetric matrix

Instructor: Mircea Nicolescu Lecture 8 CS 485 / 685 Computer Vision.

Linear Algebra Chapter 6 Linear Algebra with Applications -Gareth Williams Br. Joel Baumeyer, F.S.C.

STROUD Worked examples and exercises are in the text Programme 5: Matrices MATRICES PROGRAMME 5.

Lesson 7 Controllability & Observability Linear system 1. Analysis.

Lecture Note 1 – Linear Algebra Shuaiqiang Wang Department of CS & IS University of Jyväskylä

USSC3002 Oscillations and Waves Lecture 5 Dampened Oscillations Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive.

Reduced echelon form Matrix equations Null space Range Determinant Invertibility Similar matrices Eigenvalues Eigenvectors Diagonabilty Power.

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

Introduction The central problems of Linear Algebra are to study the properties of matrices and to investigate the solutions of systems of linear equations.

Introduction The central problems of Linear Algebra are to study the properties of matrices and to investigate the solutions of systems of linear equations.

Linear Algebra Lecture 2.

Sturm-Liouville Theory

Matrices and vector spaces

Jordan Block Under what conditions a given matrix is diagonalizable ??? Therorem 1: REMARK: Not all nxn matrices are diagonalizable A similar to.

CS485/685 Computer Vision Dr. George Bebis

Multiplicative Inverses of Matrices and Matrix Equations

Matrix Solutions to Linear Systems

MATH 374 Lecture 23 Complex Eigenvalues.

Find the area of the Triangle

Matrix Algebra and Random Vectors

Further Matrix Algebra

MATRICES Operations with Matrices Properties of Matrix Operations

ARRAY DIVISION Identity matrix Islamic University of Gaza

Matrix Definitions It is assumed you are already familiar with the terms matrix, matrix transpose, vector, row vector, column vector, unit vector, zero.

Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 – 14, Tuesday 8th November 2016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR)

Maths for Signals and Systems Linear Algebra in Engineering Lectures 16-17, Tuesday 18th November 2014 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR)

Linear Algebra Lecture 32.

3.IV. Change of Basis 3.IV.1. Changing Representations of Vectors

Linear Algebra Lecture 30.

Subject :- Applied Mathematics

Matrices - Operations INVERSE OF A MATRIX

Linear Algebra: Matrix Eigenvalue Problems – Part 2

Presentation transcript:

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN1 Lecture 28 is positive definite Similar matrices Jordan form

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN2 Positive definite means Except for Is the inverse of the symmetric positive matrix also the positive matrix? If A and B are positive definite, how about (A+B)? Now A is m by n matrix and rank(A)=n Is square and symmetric Is it positive definite? A and B are similar means For some matrix M For n by n matrices

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN3 Example : A is similar to B A is similar to B and Similar matrices have same eigenvalues has same eigenvalues as A In the same family

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN4 Eigenvector of matrix B is Bad case One small family has Big family includes Jordan form Can change to any value Best working matrix in this family Proof: Only one eigenvector Eigenvector of A

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN5 More members of family Every matrix has : 2 independent eigenvectors and 2 missing

NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN6 2 independent eigenvectors and 2 missing Jordan block and are not equal Every square matrix A is similar to a Jordan matrix J # of blocks = # of eigenvectors Good case :