M.A.P. Matrix Algebra Procedures by Mary Donovan, Adrienne Copeland, & Patrick Curry This document provides an easy to follow background and review of basic matrix definitions and algebra. Because population biology students have diverse experience and mathematic backgrounds, not all will be familiar with matrix algebra prior to taking the course. Some will need extra explanation of matrix algebra fundamentals, which will be provided by this handout. This document was designed to be simple to understand by students who have never used matrices before. It will be an asset to the course instructors because it will allow them to continue to provide a brief introduction to the topic without leaving students too far behind. This document expands on appendix 2 in the book, and puts all the information in one easy to access place, it is an intended handout to be given at the beginning of chapter 3 in Case 2000. However, given that this document was written in a very general fashion, it can be used for non population biology matrix math as well. To date, the main source of this information for the student is in the chapter 3 and appendix 2. Theses sources are deficient because they do not include a through explanation of all the procedures used in the chapter 3. For example, the text does not lay out how to calculate eigenvalues and eigenvectors. Dr. Daehler briefly covered this material in his lecture. However, without a strong background in matrix algebra, learning these can be confusing without understanding the fundamentals. This handout will provide another explanation and example to this important, yet confusing, topic. Below are the ways that this document varies from appendix 2: It starts with basic definitions, which are not described elsewhere It covers addition and subtraction (not covered at all in the book). The appendix uses the age structured population model as an example to explain matrix multiplication. But, if you are confused about matrices and confused about age structured models then you are double confused. We used basic variables and explained matrix multiplication with constants, vectors, and matrix by matrix. We provide examples of other matrix functions including: o determinant o inverse o transpose o and trace We describe eigenvalues and eigenvectors and give examples on how to solve for them.
A review of MATRIX ALGEBRA DEFINITIONS: A matrix is a collection of numbers arranged into a fixed number of rows and columns. Each number in a matrix is called an element. A vector is a collection of numbers arranged into one column or one row. Note: Bold text indicates matrix notation or Note: uppercase indicates a matrix lowercase indicates a vector M is a matrix with elements, a, b, c, and d v is a vector with elements x, and y The dimensions of a matrix are the number of rows and columns of the matrix. The dimensions of this matrix are (3 x 2), where there are 3 rows, and 2 columns. The identity matrix is a matrix that equivalent to 1, with ones in the main diagonal and zeros elsewhere. The dimensions of the identity matrix is (n x n), or a square matrix with the same number of rows as columns. 1 0 0 0 1 0 0 0 1 Main Diagonal 1
ALGEBRA: Addition and Subtraction Matrices may be added or subtracted only if they have the same dimensions, both the same number of rows and the same number of columns. Multiplication Matrices may be added or subtracted only if they have the number of columns as rows. Matrix multiplied by a constant: Matrix multiplied by a vector: Matrix multiplied by a matrix: 2
Other Matrix functions: The determinant of a matrix: det The determinant of a 2x2 matrix gives the area; the determinant of a 3x3 matrix gives the volume of the shape defined by the rows of the matrix. The determinant can tell you if the matrix can be inverted. If the deta=0 than there is no inverse. The inverse of a matrix: 1 det The transpose of a matrix: transpose of or T T T The transpose of a matrix is where the columns become rows and the rows become columns through the reflection of A over its main diagonal. The trace of a matrix: The trace is the sum of the elements of the main diagonal tr(a) = tr(a T ) tr 3
Eigenvalues, eigenvectors: The matrix remains proportional to the original vector after being multiplied by the eigenvector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector changes when multiplied by the matrix. An n x n matrix may have up to n eigenvectors with an eigenvalue associated with it. where λ is the eigenvalue of A corresponding to the eigenvector x. Solving for the eigenvalues: The characteristic polynomial: det 0 0 1 0 1 λ 0 0 λ det det Example: det 0 2 8 0 1 1 0 0 1 λ 0 0 λ Set each parenthesis equal to zero and solve for the eigenvalues. det det 2 8 1 2 8 28 4 2 4 2 Remember F.O.I.L.? The dominant eigenvalue is the eigenvalue with the largest absolute value. 4
Solving for the eigenvectors: 1 0 2 0 solve for the relationship between and 0 If this result is confusing review how to multiply a matrix by a vector (explained above) using the first result, arbitrarily set 1 to get this gives us the eigenvector 1 Example: 2 8 0 1 1 0 0 1 4 This was the dominant eigenvalue from example above 24 8 1 0 4 1 2 0 0 0 24 8 arbitrarily set 1 0.25 1 0.25 This is the dominant eigenvector because it was calculated from the dominant eigenvalue in example above 5