vectors - Eigenvalues and Eigenvectors
Solving Linear Equations
54 © 2017 University of the West Indies Open Campus UNITU 5 Vectors – Eigenvalues and Eigenvector Unit Overview In the previous Unit, we covered matrices by looking at the inverse and symmetric matrices. In this Unit, we are going to extend our discussion by looking at some other dimensions of matrices. This will allow us to expand our knowledge of vectors by providing a formal definition, and explore some of the properties of vectors. In addition, we will also explore diagonalisation and orthogonal diagonalisation of matrices in this Unit. We will utilise videos and additional readings to provide examples and facilitate further understanding of the topic. This Unit is divided into two sessions. In Session 5.1 we will be analysing vectors by looking at properties and some of the mathematical operations such as addition, subtraction and multiplication of vectors. Session 5.2 allows us to further discuss diagonalisation of matrices. We will start by looking at eigenvalues and eigenvectors and then outline the steps that are needed to diagonalise a matrix. Unit Objectives By the end of this Unit you will be able to: 1. Define vectors. 2. Define diagonalisation of matrices. 3. Perform orthogonal diagonalisation. 4. Analyse eigenvalues and eigenvectors. 55 ECON2015 Mathematical Methods of Economics I – UNIT 5 This Unit is divided into two sessions as follows: Session 5.1: Introduction, Characteristic Vectors, Diagonalisation Session 5.2: Orthogonal Diagonalisation, Some Properties of Eigenvalues and Eigenvectors Readings and Resources Note to Students: Sometimes hyperlinks to resources may not open when clicked. If any link fails to open, please copy and paste the link in your browser to view/ download the resource. Required Readings Kuttler, K. (2012). Elementary Linear Algebra,Chapter 12. The Saylor Foundation. Available at https://www.saylor.org/site/wp-content/uploads/2012/04/ Elementary-Linear-Algebra-4-26-12-Kuttler-OTC.pdf Strang, G. (2010). Linear Algebra. [Spring, 2010]. (Massachusetts Institute of Technology: MIT Open CouseWare). Available at https://ocw.mit.edu/ courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures Required Video DLBmaths. (2012). Diagonalisation of a 2x2 matrix. [YouTube]. Available at https://www.youtube.com/watch?v=INfPkT9EkhE https://www.saylor.org/site/wp-content/uploads/2012/04/Elementary-Linear-Algebra-4-26-12-Kuttler-OTC.pdf https://www.saylor.org/site/wp-content/uploads/2012/04/Elementary-Linear-Algebra-4-26-12-Kuttler-OTC.pdf https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures https://www.youtube.com/watch?v=INfPkT9EkhE 56 ECON2015 Mathematical Methods of Economics I – UNIT 5 SeesionS 5.1 Nntroduction, Characteristic Vectors, Diagonalisation Introduction We worked in the last unit with matrices, and to some extent vectors. In this session, we will delve further into the properties of vectors. We will provide a formal definition of vectors and then move into properties. We will also be reviewing the operations of matrices. These will include addition, subtraction and multiplication. We will also be doing a brief introduction on diagonalisation of matrices. This will give us an introduction to eigenvalues and eigenvectors, which will offer an introduction for the next session. Vectors A vector is an object that has both a magnitude and a direction. Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of the vector, and with an arrow indicating the direction. The direction of the vector is from its tail to its head. Properties According to Math Insight (2017), a vector has several properties. Some of the main ones are as follows: 1. Two vectors are the same if they have the same magnitude and direction. This means that if we take a vector and translate it to a new position (without rotating it), then the vector we obtain at the end of this process is the same vector we had in the beginning. 2. We denote vectors using arrows as in or , or they use other markings. Also, each vector has a magnitude and a direction. 3. Moving the vector around does not change the vector, as the position of the vector does not affect the magnitude or the direction. But if you stretch or turn http://mathinsight.org/definition/magnitude_vector 57 ECON2015 Mathematical Methods of Economics I – UNIT 5 the vector by moving just its head or its tail, the magnitude or direction will change. 4. There is one important exception to vectors having a direction. The zero vector, is the vector of zero length. Since it has no length, it is not pointing in any particular direction. Operation of Vectors We can define a number of operations on vectors geometrically without reference to any coordinate system. Here we can look at the addition, subtraction and multiplication of vectors Addition of Vectors Given two vectors and , we form their sum a + b, as follows. We translate the vector until its tail coincides with the head of . (Recall such translation does not change a vector.) Then, the directed line segment from the tail of to the head of is the vector a + b. This is captured in the diagram below. FsgurS 5.1-Sum of and Addition of vectors satisfies two important properties. 1. The commutative law, which states the order of addition doesn’t matter: a+b = b+a. 2. The associative law, which states that the sum of three vectors does not depend on which pair of vectors is added first: (a+b)+c=a+(b+c). Subtraction of Vectors Before we define subtraction, we define the vector , which is the opposite of . This vector is the vector with the same magnitude as but that is pointed in the opposite direction. Therefore, we define the subtraction of vectors as the opposite of the addition rule above: b−a=b+(−a). http://mathinsight.org/zero_vector 58 ECON2015 Mathematical Methods of Economics I – UNIT 5 Multiplication There are several techniques used for the multiplication of vectors. These, according to the Physics Hypertextbook (n.d.), are as follows: • Dot product — also known as the “scalar product”, is an operation that takes two vectors and returns a scalar quantity. The dot product of two vectors can be defined as the product of the magnitude of the two vectors and the cosine of the angle between the two vectors. The symbol used to represent this operation is a small dot at middle height (·), which is where the name “dot product” comes from. Since this product has magnitude only, it is also known as the scalar product. A · B = AB cos θ The dot product is distributive A · (B + C) = A · B + A · C • Cross product — also known as the “vector product”, an operation on two vectors that results in another vector. The cross product of two vectors in 3-space is defined as the vector perpendicular to the plane determined by the two vectors whose magnitude is the product of the magnitude of the two vectors and the sine of the angle between the two vectors. So, if n is the unit vector perpendicular to the plane determined by vectors A and B, A × B = ||A|| ||B|| sin θ n. Introduction to Diagonalisation of a Matrix Matrix diagonalisation is the process of taking a square matrix and converting it into a new matrix that shares the same fundamental properties of the underlying matrix. Matrix diagonalisation is equivalent to transforming the underlying system of equations into a special set of coordinate axes in which the matrix takes a specific form. Diagonalising a matrix is also equivalent to finding the matrix’s eigenvalues, which turn out to be precisely the entries of the diagonalised matrix. Similarly, the eigenvectors make up the new set of axes corresponding to the diagonal matrix. An eigenvalue is a special set of scalar associated with linear transformation while an eigenvector or characteristic vector of a linear transformation is a non-zero vector whose direction does not change when that linear transformation is applied to it. The remarkable relationship between a diagonalised matrix, eigenvalues, and eigenvectors follows from the beautiful mathematical identity. Inherent to diagonalisation are eigenvalues and eigenvectors. https://en.wikipedia.org/wiki/Dot_product https://en.wikipedia.org/wiki/Scalar_(mathematics) https://en.wikipedia.org/wiki/Cross_product https://en.wikipedia.org/wiki/Euclidean_vector http://mathworld.wolfram.com/SquareMatrix.html http://mathworld.wolfram.com/Matrix.html http://mathworld.wolfram.com/Eigenvalue.html http://mathworld.wolfram.com/Matrix.html http://mathworld.wolfram.com/Eigenvector.html http://mathworld.wolfram.com/DiagonalMatrix.html https://en.wikipedia.org/wiki/Linear_map https://en.wikipedia.org/wiki/Vector_space http://mathworld.wolfram.com/Eigenvalue.html http://mathworld.wolfram.com/Eigenvector.html 59 ECON2015 Mathematical Methods of Economics I – UNIT 5 Let us take a further look at these concepts. Reading Kuttler (2012) provides a basic definition of eigenvalues and eigenvectors. It starts by providing a basic definition of the two concepts and some examples of how to derive the two concepts. This will set the stage for the next session where we will use eigenvalues and eigenvectors to address a special type of matrix operations. Kuttler, K. (2012). Elementary Linear Algebra (1st ed.). The Saylor Foundation. Chapter 12, pg 216-220. Available at u https://www.saylor.org/site/wp-content/uploads/2012/04/Elementary- Linear-Algebra-4-26-12-Kuttler-OTC.pdf Examples of Diagonalisation of Matrix Play Video In this video, we will see a step-by-step example of how to diagonalise a matrix. DLBmaths (2012) is a useful follow up of the Kuttler (2012) reading. The video looks at the diagonalisation of a 2x2 matrix, making reference to eigenvectors and eigenvalues from the resource above. DLBmaths. (2012). Diagonalisation of a 2x2 matrix.[YouTube]. Available at u ttps://www.youtube.com/watch?v=INfPkT9EkhE LEARNING ACTIVITY 5.1 Based on the information learned in this session, create two vectors: call one vector and the other vector Find 1. A+B 2. 2A-B 3. Prove that the dot product is distributive Post your answers to the Discussion forum and comment on at least two posts by classmates. https://www.youtube.com/watch?v=INfPkT9EkhE 60 ECON2015 Mathematical Methods of Economics I – UNIT 5 Session Summary In Session 5.1, we provided a formal definition of vectors. We also looked at the addition, subtraction and multiplication of vectors. Examples were provided to substantiate the theory. Additionally, we had a brief introduction to diagonalisation of matrix by looking at eigenvalues and eigenvectors. This will serve as an introduction to the next session, which builds on the definition of the two concepts. 61 ECON2015 Mathematical Methods of Economics I – UNIT 5 SeesionS 5.2 Orthogonal Diagonalisation, Some Properties of Eigenvalues and Eigenvectors Introduction In this session, we will be building on the concept of diagonalisation by introducing some additional properties. We will be doing an analysis of eigenvalues and eigenvectors which will set the background for orthorgonal diagonalisation. The main properties of this process will be discussed and examples will be illustrated using a video. This video will serve to provide step-by-step instructions needed to orthogonally diagonalise a matrix. An activity is also provided to facilitate application of the concepts covered. Eigenvalues and Eigenvectors As discussed in the previous session, if we have an equation, for example: A·v=λ·v where A is an n-by-n matrix, v is a non-zero n-by-1 vector, and λ is a scalar (which may be either real or complex). Any value of λ for which this equation has a solution is known as an eigenvalue of the matrix A. It is sometimes also called the characteristic value.