Matrix Algebra 2.1 MATRIX OPERATIONS Pearson Education, Inc.

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1 2 Matrix Algebra 2.1 MATRIX OPERATIONS

2 MATRIX OPERATIONS mn If A is an matrix that is, a matrix with m rows and n columns then the scalar entry in the ith row and jth column of A is denoted by a ij and is called the (i, j)-entry of A. See the figure below. Each column of A is a list of m real numbers, which identifies a vector in R m. Slide 2.1-2

3 MATRIX OPERATIONS The columns are denoted by a 1,, a n, and the matrix A is written as A a a a 1 2 n The number a ij is the ith entry (from the to) of the jth column vector a j. mn The diagonal entries in an matrix are a 11, a 22, a 33,, and they form the main diagonal of A. A diagonal matrix is a sequence nondiagonal entries are zero. nn An examle is the identity matrix, I n.. nm A a ij matrix whose Slide 2.1-3

4 SUMS AND SCALAR MULTIPLES An mn matrix whose entries are all zero is a zero matrix and is written as 0. The two matrices are equal if they have the same size (i.e., the same number of rows and the same number of columns) and if their corresonding columns are equal, which amounts to saying that their corresonding entries are equal. mn A B If A and B are matrices, then the sum is the mn matrix whose columns are the sums of the corresonding columns in A and B. Slide 2.1-4

5 SUMS AND SCALAR MULTIPLES Since vector addition of the columns is done entrywise, each entry in A B is the sum of the corresonding entries in A and B. The sum same size. A B Examle 1: Let C is defined only when A and B are the A, B, A B A C and. Find and. Slide 2.1-5

6 SUMS AND SCALAR MULTIPLES AB A C Solution: but is not defined because A and C have different sizes. If r is a scalar and A is a matrix, then the scalar multile ra is the matrix whose columns are r times the corresonding columns in A. Theorem 1: Let A, B, and C be matrices of the same size, and let r and s be scalars. a. A B B A Slide 2.1-6

7 SUMS AND SCALAR MULTIPLES b. c. d. e. f. ( A B) C A ( B C) A0 A r( A B) ra rb ( r s) A ra sa r( sa) ( rs) A Each quantity in Theorem 1 is verified by showing that the matrix on the left side has the same size as the matrix on the right and that corresonding columns are equal. Slide 2.1-7

8 MATRIX MULTIPLICATION When a matrix B multilies a vector x, it transforms x into the vector Bx. If this vector is then multilied in turn by a matrix A, the resulting vector is A (Bx). See the Fig. below. Thus A (Bx) is roduced from x by a comosition of maings the linear transformations. Slide 2.1-8

9 MATRIX MULTIPLICATION Our goal is to reresent this comosite maing as multilication by a single matrix, denoted by AB, so that. See the figure below. AB ( x)=(ab)x mn n If A is, B is, and x is in R, denote the columns of B by b 1,, b and the entries in x by x 1,, x. Slide 2.1-9

10 MATRIX MULTIPLICATION Then Bx x b... x b 1 1 By the linearity of multilication by A, A( Bx) A( x b )... A( x b ) 1 1 x Ab... x Ab 1 1 The vector A (Bx) is a linear combination of the vectors Ab 1,, Ab, using the entries in x as weights. In matrix notation, this linear combination is written as. A( Bx) Ab Ab Ab x 1 2 Slide

11 MATRIX MULTIPLICATION Thus multilication by A b Ab Ab 1 2 transforms x into A (Bx). mn n Definition: If A is an matrix, and if B is an matrix with columns b 1,, b, then the roduct AB is the matrix whose columns are Ab 1,, Ab. m That is, AB A b b b b b b 1 2 A A A 1 2 Multilication of matrices corresonds to comosition of linear transformations. Slide

12 MATRIX MULTIPLICATION Examle 2: Comute AB, where B A and Solution: Write B b b b 1 2 3, and comute: Slide

13 Ab 1 MATRIX MULTIPLICATION , Ab, Ab Then AB A b b b Ab 1 Ab 2 Ab 3 Slide

14 MATRIX MULTIPLICATION Each column of AB is a linear combination of the columns of A using weights from the corresonding column of B. Row column rule for comuting AB If a roduct AB is defined, then the entry in row i and column j of AB is the sum of the roducts of corresonding entries from row i of A and column j of B. If (AB) ij denotes the (i, j)-entry in AB, and if A is an mnmatrix, then. ( AB) a b... a b ij i1 1j in nj Slide

15 PROPERTIES OF MATRIX MULTIPLICATION mn Theorem 2: Let A be an matrix, and let B and C have sizes for which the indicated sums and roducts are defined. A( BC) ( AB) C a. (associative law of multilication) A( B C) AB AC ( B C) A BA CA r( AB) ( ra) B A( rb) I A A AI b. (left distributive law) c. (right distributive law) d. for any scalar r e. (identity for matrix m n multilication) Slide

16 PROPERTIES OF MATRIX MULTIPLICATION Proof: Proerty (a) follows from the fact that matrix multilication corresonds to comosition of linear transformations (which are functions), and it is known that the comosition of functions is associative. Let C c c 1 By the definition of matrix multilication, BC Bc Bc 1 A( BC) A( Bc ) A( Bc ) 1 Slide

17 PROPERTIES OF MATRIX MULTIPLICATION The definition of AB makes x, so A( Bx) ( AB)x for all A( BC) ( AB)c ( AB)c ( AB) C 1 The left-to-right order in roducts is critical because AB and BA are usually not the same. Because the columns of AB are linear combinations of the columns of A, whereas the columns of BA are constructed from the columns of B. The osition of the factors in the roduct AB is emhasized by saying that A is right-multilied by B or that B is left-multilied by A. Slide

18 PROPERTIES OF MATRIX MULTIPLICATION AB BA If, we say that A and B commute with one another. Warnings: AB BA 1. In general,. 2. The cancellation laws do not hold for matrix multilication. That is, if AB AC, then it is not true in general that. 3. If a roduct AB is the zero matrix, you cannot conclude in general that either or. B C A 0 B 0 Slide

19 POWERS OF A MATRIX nn If A is an matrix and if k is a ositive integer, then A k denotes the roduct of k coies of A: k A A A If A is nonzero and if x is in R n, then A k x is the result of left-multilying x by A reeatedly k times. k If k 0, then A 0 x should be x itself. Thus A 0 is interreted as the identity matrix. Slide

20 THE TRANSPOSE OF A MATRIX mn Given an matrix A, the transose of A is the nm matrix, denoted by A T, whose columns are formed from the corresonding rows of A. Theorem 3: Let A and B denote matrices whose sizes are aroriate for the following sums and roducts. a. b. c. For any scalar r, d. ( T T A ) A ( A B) T A T B T ( ra) T ra T ( AB) T B T A T Slide

21 THE TRANSPOSE OF A MATRIX The transose of a roduct of matrices equals the roduct of their transoses in the reverse order. Slide