Matrix Algebra: Definitions and Basic Operations
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1 Section 4 Matrix Algebra: Definitions and Basic Operations Definitions Analyzing economic models often involve working with large sets of linear equations. Matrix algebra provides a set of tools for dealing with such objects. A matrix is a rectangular collection of numbers a11 a12 a1n a21 a22 a 2n A. am1 am2 amn The number of rows m need not be equal to the number of columns n. A matrix with m rows and n columns is said to have order (m,n) or dimension (m,n), or we simply call it a ( m n) matrix. The number that appears in the (i, j)th position is called the (i, j)th element or the (i, j)th entry of the matrix. If m n, the matrix is a square matrix. If m 1 and n 1, it is called a row vector. If m 1 and n 1, we have a column vector. If mn 1, then we have a scalar. The elements of a vector are often called the components of the vector. b1 b 2 A row vector c c1 c2 cn, a column vector b. bm By simply stating that b is a vector we will usually mean that b is a column vector, but you need to be aware whether a row or column vector is being referred to. Matrices are often written in unitalicized bold uppercase letters, vectors in unitalicized bold lowercase letters. It is often convenient to write a matrix as A ( a ), mn and often convenient to refer to the (i, j)th element of A using A. There is variation in notation from author to author, so be careful in your reading. Two matrices of the same dimension m n are said to be equal if all of their corresponding elements are equal, i.e., A B [ A] [ B ] i 1,2,..., m, j 1,2,..., n. Matrices of different dimensions cannot be equal. 4-1
2 Basic Operations (Addition, Scalar Multiplication, Subtraction, Transpose) Addition Let A ( a ) and B ( b ) be two arbitrary ( m n) matrices. Define mn mn A B ( a b ) mn, i.e., addition of matrices is defined to be element by element addition Matrices being added together obviously must have the same dimensions. It should also be obvious that A B B A ( A B) C A ( B C ) This means that as far as addition is concerned, we can manipulate matrices in the same way we manipulate ordinary numbers (as long as they have the same dimensions) Scalar Multiplication Let A be a ( m n) matrix, and be a scalar. Then define A ( a ) m n i.e., the product of a scalar and a matrix is defined to be the multiplication of each element of the matrix by the scalar. a11 a12 ba11 ba12 b a a ba ba a31 a32 ba31 ba32 We can use scalar multiplication to define matrix subtraction. Let A and B be ( m n) matrices. Then A B A ( 1) B
3 Transpose An important operator is the transpose. When we transpose a matrix, we write its rows as its column, and its columns as its rows. For example, denoting the transpose of A by A we have Put more succinctly, [ A ] [ A ] ji. Note that transposes are often denoted using A instead of A. One application of the transpose operator is in defining symmetric matrices. A symmetric matrix is defined as one where A A. Exercises 1. Let 7 13 A 4 4. What is the dimension of A? What is [ A ] 12? What is [ A ] 31? Suppose A ( a ) 24 where a i j. Write out the matrix in full. 3. Write out in full the matrix (i) ( a ) 4 4 where a 1 when i j, 0 otherwise. (ii) ( a ) 4 4 where a 0 if i j. (Fill in the rest of the entries * ) (iii) ( a ) 55 where a 0 when i j. (Fill in the rest of the entries with * ) (vi) ( a ) 55 where a 0 when i j. (Fill in the rest of the entries with * ) These are all square matrices. Matrices (i) and (ii) are called diagonal matrices. Matrix (iii) is a lower triangular matrix, and (iv) is an upper triangular matrix (so we have in (iii) and (iv) matrices that are square and triangular!) 4. Give an example of a (4 4) matrix such that [ A] [ A ]. ji 5. If u 2v u v, what is u and v?
4 6. Let v 1, v 2, v 3, v 4 represent cities, and suppose there are one-way flights from v 1 to v 2 and v 3, from v 2 to v 3 and v 4, and two-way flights between v 1 and v 4. Write out a matrix A such that [ A ] 1 if there is a flight from v i to v j, and zero otherwise. 7. What is the dimension of the matrix ? Let A and B 0 0. Is A B? 0 0 Matrices with all zero entries are called zero matrices, and written 0 mn,, or 0 n if square, or simply 0 if the dimensions can be easily obtained from context. 9. If 1 u , what is u, v, and w? v w If 3 4 2A 2 8, what is A? If B , what is B? Which of the following matrices are symmetric? (a) b b 3 1 (b) b (c) (d) b b 3 1 (e)
5 12. True or False? (a) Symmetric matrices must be square. (b) A scalar is symmetric. (c) If A is symmetric, then A is symmetric. (d) The sum of symmetric matrices is symmetric. (e) If ( A ) A, then A is symmetric. 13. (a) Find A and B if they satisfy simultaenously AB A2B (b) If A B C and 3A 2B 0 simultaneously, find A and B in terms of C. 4-5
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