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Unformatted text preview: ___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University  Ansan 31 Chapter 3. Matrices ♣ Arithmetic on the matrices based on its algebraic structure ♣ Matrix as a transformation (or function) 1. Matrix operations A matrix, A , is a rectangular array of elements (or entries). The size (or dimension ) of a matrix is the numbers of rows and columns of it and we usually denote it as ( ) n m × ∈ A , where m and n refer to the number of rows and columns, respectively. Some important class of matrices 1. row vector, ( ) n × ∈ 1 a 2. column vector, ( ) 1 × ∈ m b 3. square matrix, ( ) m m × ∈ A 4. diagonal matrix, ( ) ( ) m m a a a diag m × ∈ = L 2 1 D 5. identity matrix, ( ) ( ) m m diag m × ∈ = 1 1 1 L I (1) Equality Two matrices are equal if they have the same size and the corresponding entries are equal; i.e . if n m ij a × = ] [ A and s r ij b × = ] [ B , then B A = , iff m=r , n=s , and a ij = b ij for all i and j . (2) Matrix addition We define matrix addition componentwise. If ] [ ij a = A and ] [ ij b = B are ( ) n m × matrices, their sum B A + is the ( ) n m × matrix obtained by adding the corresponding entries; i.e. n m ij ij b a × + = + ] [ B A (3) Scalar multiple n m ij ca c × = ] [ A (4) Matrix multiplication The product of two matrices n m ij a × = ] [ A and r n ij b × = ] [ B is an ( ) r m × matrix AB C = whose entries are computed as nj in j i j i ij b a b a b a c + + + = L 2 2 1 1 ① Let e i be a ) 1 ( m × standard unit row vector, then A e i is the i th row of A . ② Let e j be a ) 1 ( × n standard unit column vector, then j Ae is the j th column of A . ___________________________________________________________ Linear Algebra: Spring 2007 __________________________________________________________________________________ Proprietary of SAS Lab., Hanyang University  Ansan 32 ③ Put [ ] cn c c rm r r a a a a a a A L M 2 1 2 1 = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = and [ ] cr c c rn r r b b b b b b B L M 2 1 2 1 = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = , where ( ) 1 × ∈ n ci b is the i th column of B , ( ) r ri × ∈ 1 b is the i th row of B , ( ) n ri × ∈ 1 a is the i th row of A, and ( ) 1 × ∈ m ci a is the i th column of A (These are known as the partition of matrices). Then [ ] rn cn r c r c rm r r cr c c b a b a b a B a B a B a Ab Ab Ab AB + + + = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = = L M L 2 2 1 1 2 1 2 1 ④ Similarly, if [ ] ( ) m c c c m × ∈ = 1 2 1 L c and ( ) 1 2 1 × ∈ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = n d d d n M d , then rm m r r a c a c c + + + = L 2 2 1 1 a cA and cn n c c d d d a a a Ad + + + = L 2 2 1 1 ....
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 Spring '07
 Chung
 Linear Algebra, Algebra, Matrices, Matrix Operations

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