Chapter2 - Chapter 2 Matrices Section 2.1 Introduction to...

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Chapter 2 Matrices Section 2.1 Introduction to Matrices Definition 2.1.1 A matrix (plural matrices ) is a rectangular array of numbers. The numbers in the array are called entries in the matrix. The size of a matrix is given by m × n where m is the number of rows and n is the number of columns. The ( i, j )- entry of a matrix is the number which is in the i th row and j th column of the matrix. Example 2.1.2 1. 1 2 3 4 0 - 1 is a 3 × 2 matrix. The (1,2)-entry of the matrix is 2 and the (3,1)-entry is 0. 2. ( 2 1 0 ) is a 1 × 3 matrix. 3. 2 3 . 1 - 2 3 1 2 0 0 π 0 is a 3 × 3 matrix. 4. ( 4 ) is a 1 × 1 matrix. 5. 1 1 2 is a 3 × 1 matrix. Definition 2.1.3 A column matrix (or a column vector ) is a matrix with only one column. A row matrix (or a row vector ) is a matrix with only one row.
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2 Chapter 2. Matrices Example 2.1.4 The matrix in Example 2.1.2.5 is a column matrix and the matrix in Example 2.1.2.2 is a row matrix. The matrix in Example 2.1.2.4 is both a column and row matrix. Notation 2.1.5 In general, an m × n matrix can be 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 . or simply A = ( a ij ) m × n where a ij is the ( i, j )-entry of A . Sometimes, if the size of the matrix is already known, we may just write A = ( a ij ). Example 2.1.6 Write down the following matrices explicitly: 1. A = ( a ij ) 2 × 3 where a ij = i + j . 2. B = ( b ij ) 3 × 2 where b ij = 1 if i + j is even - 1 if i + j is odd. Answers 1. A = 2 3 4 3 4 5 ; 2. B = 1 - 1 - 1 1 1 - 1 . Definition 2.1.7 The following are some special types of matrices: 1. A matrix is called a square matrix if it has the same number of rows and columns. In particular, an n × n square matrix is called a square matrix of order n . 2. Given a square matrix A = ( a ij ) of order n , the diagonal of A is the sequence of entries a 11 , a 22 , . . . , a nn . The entries a ii are called the diagonal entries while a ij , i 6 = j , are called non-diagonal entries . A square matrix is called a diagonal matrix if all its non-diagonal entries are zero, i.e. A = ( a ij ) n × n is diagonal a ij = 0 whenever i 6 = j. 3. A diagonal matrix is called a scalar matrix if all its diagonal entries are the same, i.e. A = ( a ij ) n × n is scalar a ij = 0 if i 6 = j c if i = j for a constant c .
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Section 2.1. Introduction to Matrices 3 4. A diagonal matrix is called an identity matrix if all its diagonal entries are 1. We use I n to denote the identity matrix of order n . Sometimes we write I instead of I n when there is no danger of confusion. 5. A matrix with all entries equal to zero is called a zero matrix . We denote the m × n zero matrix by 0 m × n , or simply 0 . 6. A square matrix ( a ij ) is called symmetric if a ij = a ji for all i, j . 7. A square matrix ( a ij ) is called upper triangular if a ij = 0 whenever i > j ; and a square matrix ( a ij ) is called lower triangular if a ij = 0 whenever i < j . Both upper and lower triangular matrices are called triangular matrices .
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