Eigenvector
An eigenvector is a vector that remains in the same direction after a linear transformation is applied to it. In the context of technology and engineering, eigenvectors are used in various applications such as image processing, data compression, and machine learning algorithms. They are important for understanding the behavior of systems and for solving complex mathematical problems.
4 Key excerpts on "Eigenvector"
- Michael B. Miller(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...Still, the concept of a vector can be useful when working through the problems. For our purposes, whether we imagine a collection of data to represent a point or a vector, the math will be the same. 2 In physics and other fields, the inner product of two vectors is often denoted not with a dot, but with pointy brackets. Under this convention, the inner product of a and b would be denoted < a, b >. The term dot product can be applied to any ordered collection of numbers, not just vectors, while an inner product is defined relative to a vector space. For our purposes, when talking about vectors, the terms can be used interchangeably. 3 We have not formally introduced the concept of eigenvalues and Eigenvectors. For the reader familiar with these concepts, the columns of P are the Eigenvectors of Σ, and the entries along the diagonal of D are the corresponding eigenvalues. For small matrices, it is possible to calculate the Eigenvectors and eigenvalues by hand. In practice, as with matrix inversion, for large matrices this step almost always involves the use of commercial software packages....
eBook - ePub- Michael Harrison, Patrick Waldron(Authors)
- 2011(Publication Date)
- Routledge(Publisher)
...Eigenvalues and Eigenvectors DOI: 10.4324/9780203829998-5 3.1 Introduction Some of the basic ideas and issues encountered in the previous chapters are often covered in an introductory course in mathematics for economics and finance. The fundamental ideas of eigenvalues and Eigenvectors and the associated theorems introduced in this chapter are probably not. Many readers are therefore likely to be encountering these concepts for the first time. Hence this chapter begins by providing definitions and illustrations of eigenvalues and Eigenvectors, and explaining how they can be calculated. It goes on to examine some of the uses of these concepts and to establish a number of theorems relating to them that will be useful when we return to the detailed analysis of our various applications. 3.2 Definitions and illustration Eigenvalues and Eigenvectors arise in determining solutions to equations of the form A x = λ x (3.1) where A is an n × n matrix, x is a non-zero n -vector and λ is a scalar, and where the solution is for λ and x, given A. We shall call equations like (3.1) eigenequations. The scalar λ is called an eigenvalue of A, while x is known as an Eigenvector of A associated with λ. Sometimes the value, λ, and the vector, x, are called the proper, characteristic or latent value and vector. Consider the matrix A = [ 2 0 8 − 2 ] and the vector x = [ 1 2 ]. Since A x = [ 2 0 8 − 2 ] [ 1 2 ] = [ 2 4 ] = 2 x (3.2) λ = 2 is an eigenvalue of A and x is an associated Eigenvector. It is easy to check, by substituting into the eigenequation (3.1), that another Eigenvector of A associated with λ = 2 is [−1 − 2] ┬. Likewise, another eigenvalue of A is −2, which has associated with it Eigenvectors such as [0 1] ┬ and [0 − 1] ┬. Thus, for a given λ, we note that there are multiple associated Eigenvectors. For given λ and x, A may be viewed as the matrix that, by pre-multiplication, changes all of the elements of x by the same proportion, λ...
eBook - ePub- Adil H. Mouhammed(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Chapter One Vectors and Matrices Vectors and matrices are very important tools in economic analysis. This chapter outlines the algebra of vectors. Students will not only be able to add and subtract vectors but also to multiply vectors. For economic modeling, linear independence is indispensable in that economic models must have unique solutions, and linear independence provides these solutions. Similarly, linear and convex combinations are also important for economics. An extension of vector analysis is matrix algebra. The types of matrices and matrix operations are outlined in this chapter. A system of equations is solved by using matrix methods such as the inverse method, Cramer’s rule and the Gauss-Jordan method. Eigenvalues and vectors are explained and used for diagonalizing matrices. Finally, various economic applications of matrices are provided. Vectors A vector is a symbol used to refer to a set of variables or coefficients. For example, Y = (y 1, y 2, y 3, …, y n), where Y is used to represent n-variables. Vector A can be represented by A = (a 1, a 2, a 3, …, a n), where A contains a set of coefficients denoted by a’s. Also, vector Y can also be a set of real integer numbers such as Y = (3, 4, 6, 8). Vector Y has a dimension as well. The dimension of a vector is the total number of elements (components) in that vector. For example, the above vector, Y, has four components and hence it is said to be of dimension four. Very compactly, a vector such as Y, where Y = (5, 6, 7), has one row and three columns. Accordingly, it is said to be a vector of dimension 1 by 3 or Y 1 × 3 or Y 13, where 1 and 3 indicate number of rows and columns, respectively. In general, Y m × n is said to be a vector of dimension m × n: with m rows and n columns. Example: A = (3 3), B = (5 6 7), C = [ 4 6 7 ] All these vectors have different dimensions: A is 1 × 2, B is 1 × 3, and A is 3 × 1...
eBook - ePub- Adil H. Mouhammed(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...CHAPTER ONE Vectors and Matrices Vectors and matrices are very important tools in economic analysis. This chapter outlines the algebra of vectors. Students will not only be able to add and subtract vectors but also to multiply vectors. For economic modeling, linear independence is indispensable in that economic models must have unique solutions, and linear independence provides these solutions. Similarly, linear and convex combinations are also important for economics. An extension of vector analysis is matrix algebra. The types of matrices and matrix operations are outlined in this chapter. A system of equations is solved by using matrix methods such as the inverse method, Cramer’s rule and the Gauss-Jordan method. Eigenvalues and vectors are explained and used for diagonalizing matrices. Finally, various economic applications of matrices are provided. Vectors A vector is a symbol used to refer to a set of variables or coefficients. For example, Y = (y 1, y 2, y 3,..., y n), where Y is used to represent n-variables. Vector A can be represented by A = (a 1, a 2, a 3,..., a n), where A contains a set of coefficients denoted by a’s. Also, vector Y can also be a set of real integer numbers such as Y = (3, 4, 6, 8). Vector Y has a dimension as well. The dimension of a vector is the total number of elements (components) in that vector. For example, the above vector, Y, has four components and hence it is said to be of dimension four. Very compactly, a vector such as Y, where Y = (5, 6, 7), has one row and three columns. Accordingly, it is said to be a vector of dimension 1 by 3 or Y 1 × 3 or Y 1 3, where 1 and 3 indicate number of rows and columns, respectively. In general, Y m x n is said to be a vector of dimension m x n: with m rows and n columns. Example: All these vectors have different dimensions: A is 1 × 2, B is 1 × 3, and A is 3 × 1...
