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Unformatted text preview: POL502: Linear Algebra Kosuke Imai Department of Politics, Princeton University December 12, 2005 1 Matrix and System of Linear Equations Definition 1 A m n matrix A is a rectangular array of numbers with m rows and n columns and written as A = a 11 a 12 a 1 n a 21 a 22 a 2 n . . . . . . . . . . . . a m 1 a m 2 a mn a ij is called the ( i, j ) th element of A . Note that a special case of matrix is a vector where either m = 1 or n = 1. If m = 1 and n > 1, then it is called a row vector . If m > 1 and n = 1, then it is called a column vector . A vector has a nice geometric interpretation where the direction and length of the vector are determined by its elements. For example, the vector A = [1 3] has the opposite direction and twice as long as the vector B = [- 1 / 2- 3 / 2]. We will discuss vectors in more detail later in this chapter. Now, we define the basic operations of matrices. Definition 2 Let A and B be m n matrices. 1. (equality) A = B if a ij = b ij . 2. (addition) C = A + B if c ij = a ij + b ij and C is an m n matrix. 3. (scalar multiplication) Given k R , C = kA if c ij = ka ij where C is an m n matrix. 4. (product) Let C be an n l matrix. D = AC if d ij = n k =1 a ik c kj and D is an m l matrix. 5. (transpose) C = A > if c ij = a ji and C is an n m matrix. Example 1 Calculate A + 2 B > , AB , and BA using the following matrices. A = 1 2- 1 3 1 3 , B = - 2 5 4- 3 2 1 , The basic algebraic operations for matrices are as follows: Theorem 1 (Algebraic Operations of Matrices) Let A , B , C be matrices of appropriate sizes. 1. Addition: 1 (a) A + B = B + A and A + ( B + C ) = ( A + B ) + C (b) There exists a unique C such that A + C = A and we denote C = O . (c) There exists a unique C such that A + C = O and we denote C =- A . 2. Multiplication: (a) k ( lA ) = ( kl ) A , k ( A + B ) = kA + kB , ( k + l ) A = kA + lA , and A ( kB ) = k ( AB ) = ( kA ) B for any k, l R . (b) A ( BC ) = ( AB ) C . (c) ( A + B ) C = AC + BC and C ( A + B ) = CA + CB . 3. Transpose: (a) ( A > ) > = A . (b) ( A + B ) > = A > + B > . (c) ( AB ) > = B > A > . (d) ( kA ) > = kA > . Example 2 Calculate ( ABC ) > using the following matrices. A = 1 2 3- 2 1 , B = 1 2 2 3- 1 , C = 3 2 3- 1 , There are some important special types of matrices. Definition 3 Let A be an m n matrix. 1. A is called a square matrix if n = m . 2. A is called symmetric if A > = A . 3. A square matrix A is called a diagonal matrix if a ij = 0 for i 6 = j . A is called upper triangular if a ij = 0 for i > j and called lower triangular if a ij = 0 for i < j . 4. A diagonal matrix A is called an identity matrix if a ij = 1 for i = j and is denoted by I n ....
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- Spring '08