Matrices and their operations No. 1

Transcription

1 Matrices and their operations No. 1 Multiplication of 2 2 Matrices Nobuyuki TOSE October 11, 2016

2 Review 1: 2 2 Matrices 2 2 matrices A ( a 1 a 2 ) ( a1 a 2 ) ( ) a11 a 12 a 21 a 22 A 2 2 matrix is given in the following ways. (i) Combining two column vectors a 1, a 2 R 2 (ii) Combining two row vectors a 1 and a 2 (iii) Giving 2 2 components. NB a ij is used for the component of the ith row and of the jth column.

3 Review 2 Multiplication of 2-dim. vectors to 2 2 matrices A ( ) x y x a 1 + y a 2 ( ) a11 x + y a 21 ( ) x a 1 y ( ) x a 2 y ( a12 a 22 ) ( ) xa11 + ya 12 xa 21 + ya 22 Here we use the multiplication of a row vector and a column vector defined by ( ) x (α β) αx + βy y

4 Review 3: Multiplication of two 2 2 matrices Take another 2 2 matrix B ( b 1 b2 ) ( b1 b 2 ). Then ( AB (A b 1 A a b 2 ) 1 b1 a 2 b1 ) a 1 b2 a 2 b2

5 Linear Map defined by using A A Map defined by A Given a 2 2 matrix we can define a map A ( a 1 a 2 ) ( a1 a 2 ) ( ) a11 a 12, a 21 a 22 F A : R 2 R 2 ( ) ( ) s s A s a t t 1 + t a 2

6 Linearity of F A Linearity of F A F A satisfies the following basic properties called Linearity. (i) F A ( x + y) F A ( x) + F A ( y) (ii) F A (λ x) λf A ( x) (iii) F A (λ x + µ y) λf A ( x) + µf A ( y) These three propeties are identical to the following. (i) A( x + y) A x + A y (ii) A(λ x) λ(a x) (iii) A(λ x + µ y) λ(a x) + µ(a y) Moreover remark that (iii) can be easily derived from (i) and (ii). In fact A(λ x + µ y) A(λ x) + A(µ y) λ(a x) + µ(a y)

7 Proof Proof for (i) LHS A (( x1 x 2 ) + ( y1 y 2 (x 1 + y 1 ) a 1 + (x 2 + y 2 ) a 2 )) ( ) x1 + y A 1 x 2 + y 2 Proof for (ii) x 1 a 1 + y 1 a 1 + x 2 a 2 + y 2 a 2 (x 1 a 1 + x 2 a 2 ) + (y 1 a 1 + y 2 a 2 ) ( ) ( ) x1 y1 A + A RHS x 2 y 2 ( ) λx1 LHS A λx 2 (λx 1 ) a 1 + (λx 2 ) a 2 λ(x 1 a 1 ) + λ(x 2 a 2 ) λ(x 1 a 1 + x 2 a 2 ) RHS

8 Associativity Thanks to the Linearity, we can prove the following theorem about Associativity. Theorem: Associativity Given 2 2 matrices A and B. Then we have for x R 2. In fact, (AB) x A(B x) RHS A(x 1 b1 + x 2 b2 ) x 1 (A b 1 ) + x 2 (A b 2 ) ( ) (A b 1 A x1 b 2 ) (AB) x LHS x 2

9 Associativity 2 Theorem: Associativity Given 2 2 matrices A, B and C. Then (AB)C A(BC)

10 Scalar Multiplication to Matirces Scalar Multiplication to 2 2 Matrices Given a 2 2 matix A ( a 1 a 2 ) ( a1 a 2 ) ( ) a11 a 12, a 21 a 22 we define a scalar multiplication by λ to A as follows: ( ) ( ) λa1 λa11 λa λa (λ a 1 λ a 2 ) 12 λa 2 λa 21 λa 22

11 Scalar Multiplication to Matirces Theorem (i) (λa) x λ(a x) A(λ x) (ii) (λa)b λ(ab) A(λB) The proof for (i) is given as follows: (λa) x (λ a 1 λ a 2 ) x (λx 1 ) a 1 + (λx 2 ) a 2 λ(x 1 a 1 ) + λ(x 2 a 2 ) λ(x 1 a 1 + x 2 a 2 ) λ(a x) Moreover the property (ii) is derived easily from (i).

12 Other Basic Properties of Scalar Multiplication Theorem (iii) (λ + µ)a λa + µa (iv) (λµ)a λ (µa) (v) 1A A and 0A O 2 These properties can derived from the following corresponding properties for vectors. It is necessary to define the addition of matrices to understand (iii), and we put it off for a couple of weeks. (iii) (λ + µ) a λ a + µ a (iv) (λµ) a λ(µ a) (v) 1 a a and 0 a 0

13 Special Matrices O 2 Zero Matrix O 2 ( 0 0) ( ) is called the zero matrix. It satisfies the identity O 2 X O 2 follows from and XO 2 O 2 from O 2 x ( 0 0) X 0 ( x 1 x 2 ) O 2 X XO 2 O 2 ( x1 x 2 ) x x ( ) 0 0 x x 2 0

14 Special Matrices 2 I 2 Identity Matrix I 2 ( e 1 e 2 ) ( ) is called the Identity Matrix. It enjoys for a 2 2 matrix X, the identity XI 2 I 2 X X It follows from the identities that ( a b) e 1 1 a + 0 b a, ( a b) e 2 0 a + 1 b b XI 2 ( x 1 x 2 )( e 1 e 2 ) ( x 1 x 2 ) X Moreover I 2 X X from ( ) x ( e 1 e 2 ) x e y 1 + y e 2 x ( ) 1 + y 0 ( ) 0 1 ( ) x y

15 Inverse matrix We have the identity ( ) ( ) ( ) ( ) a b d b d b a b c d c a c a c d ( ad bc 0 ) 0 ad bc Cofactor Matrix ( ) a b For A, its cofactor matrix is defeined by c d Then we have the identity ( ) d b à c a Aà ÃA A I 2

16 Inverse Matrix 2 Assume that A 0. Then multiply by 1 A and get A 1 A Ã 1 A Ã A I 2 Inverse Matrix In case A 0, the Inverse Matrix of A is defined by A 1 1 ( ) A Ã 1 d b ad bc c a

17 Regularity of Matrices, Uniqueness of Inverse Regularity of Matrices A 2 2 matrix A is called regular if there exists another 2 2 matrix X satisfying AX XA I 2 In this situation X is called the inverse of A. (i) If A 0, A is regular. (ii) (Uniqueness of the inverse) Assume AX XA I 2, AY YA I 2 Then X Y. In fact, from AX I 2 multiplied by Y from the left follows Y (AX ) YI 2 Y On the other hand, Y (AX ) (YA)X I 2 X X. Accordingly X Y.

18 In case A 0 Theorem A ( ) a b Let A a 2 2 matrix with A 0. Then there exists c d v 0 satisfying A v 0. ( ) ( ) a b d c d c ( a b c d ) ( ) b a ( ) 0 0 ( ) ( ) d d (i) In case d 0 or c 0, 0 AND A c c ( ) ( ) b b (ii) In case b 0 or c 0, 0 AND A a a (iii) Not (i) AND Not (ii). Then A O

19 Equivalent conditions for regularity If A is regular, then A v 0 implies v 0. In fact by multiplying A 1 to A v 0 to get A 1 A v A 1 0 namely v I 2 v 0 Thus it follows from Theorem A that if A 0 then A is not regular. Theorem B The following (i), (ii) and (iii) are equivalent for a 2 2 matrix A. (i) A is regular. (ii) A v 0 v 0 (iii) A 0.

20 Proof for Theorem B (i) (iii) The contraposition Not (iii) Not (i) is already shown. (i) (ii) Already shown. (iii) (i) Already shown. The contraposition Not (iii) Not (ii) is given in Theorem A.

21 Addition of two 2 2 Matrices Definition Given two 2 2 matrices and A ( a 1 a 2 ) B ( b 1 b2 ) ( a1 a 2 ( b1 b 2 ) ( ) a11 a 12 a 21 a 22 ) ( ) b11 b 12 b 21 b 22 Then the 2 2 matrixa + B is defined by ( ) ( ) A + B ( a 1 + b 1 a 2 + a1 + b b 2 ) 1 a11 + b 11 a 12 + b 12 a 2 + b 2 a 21 + b 21 a 22 + b 22

22 Basic Properties (1) Basic Properties (1) (i) (A + B) + C A + (B + C) (ii) A + O 2 O 2 + A A (iii) A + B B + A (iv) λ(a + B) λa + λb (v) (λ + µ)a λa + µa (i) follows from ( a + b) + c a + ( b + c). (ii) follows from a a a. (iii) follows from a + b b + a. (v) follows from (λ + µ) a λ a + µ a. (iv) is proved as follows. LHS λ( a 1 + b 1 a 2 + b 2 ) (λ( a 1 + b 1 ) λ( a 2 + b 2 )) (λ a 1 + λ b 1 λ a 2 + λ b 2 ) (λ a 1 λ a 2 ) + (λ b 1 λ b 2 ) RHS

23 Basic Properties (2) Basic Properties (2) (vi) A(B + C) AB + AC (vii) (B + C)A BA + CA (vi) follows from A( b + c) A b + A c. In fact LHS A( b 1 + c 1 b2 + c 2 ) (A( b 1 + c 1 ) A( b 2 + c 2 )) (A b 1 + A c 1 A b 2 + A c 2 ) (A b 1 A b 2 ) + (A c 1 A c 2 ) RHS (vii) follows from the identity (B + C) a B a + C a which is derived as follows. ( ) LHS ( b 1 + c 1 a1 b2 + c 2 ) a 1 ( b 1 + c 1 ) + a 2 ( b 2 + c 2 ) a 2 (a 1 b1 + a 2 b2 ) + (a 1 c 1 + a 2 c 2 ) RHS