MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

Transcription

1 MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

2 Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11 a a 1n a 21 a a 2n a m1 a m2... a mn Notation: A = (a ij ) 1 i n,1 j m or simply A = (a ij ) if the dimensions are known.

3 Matrix algebra: linear operations Addition: two matrices of the same dimensions can be added by adding their corresponding entries. Scalar multiplication: to multiply a matrix A by a scalar r, one multiplies each entry of A by r. Zero matrix O: all entries are zeros. Negative: A is defined as ( 1)A. Subtraction: A B is defined as A + ( B). As far as the linear operations are concerned, the m n matrices can be regarded as mn-dimensional vectors.

4 Properties of linear operations (A + B) + C = A + (B + C) A + B = B + A A + O = O + A = A A + ( A) = ( A) + A = O r(sa) = (rs)a r(a + B) = ra + rb (r + s)a = ra + sa 1A = A 0A = O

5 Dot product Definition. The dot product of n-dimensional vectors x = (x 1, x 2,...,x n ) and y = (y 1, y 2,...,y n ) is a scalar n x y = x 1 y 1 + x 2 y x n y n = x k y k. k=1 The dot product is also called the scalar product.

6 Matrix multiplication The product of matrices A and B is defined if the number of columns in A matches the number of rows in B. Definition. Let A = (a ik ) be an m n matrix and B = (b kj ) be an n p matrix. The product AB is defined to be the m p matrix C = (c ij ) such that c ij = n k=1 a ikb kj for all indices i, j. That is, matrices are multiplied row by column: ( ) * ( ) * = * * * * *

7 a 11 a a 1n v 1 a 21 a a 2n A = = v 2. a m1 a m2... a mn v m b 11 b b 1p b 21 b b 2p B = = (w 1,w 2,...,w p ) b n1 b n2... b np v 1 w 1 v 1 w 2... v 1 w p v 2 w 1 v 2 w 2... v 2 w p = AB = v m w 1 v m w 2... v m w p

8 Examples. y 1 (x 1, x 2,...,x n ) y 2. = ( n k=1 x ky k ), y n y 1 y 1 x 1 y 1 x 2... y 1 x n y 2. (x 1, x 2,...,x n ) = y 2 x 1 y 2 x 2... y 2 x n y n y n x 1 y n x 2... y n x n

9 Example. ( ) ( ) = ( ) is not defined

10 System of linear equations: a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a m1 x 1 + a m2 x a mn x n = b m Matrix representation of the system: a 11 a a 1n x 1 b 1 a 21 a a 2n x = b 2. a m1 a m2... a mn x n b m

11 a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 Ax = b, a m1 x 1 + a m2 x a mn x n = b m where a 11 a a 1n x 1 b 1 a 21 a a 2n A =......, x = x 2., b = b 2.. a m1 a m2... a mn x n b m

12 Properties of matrix multiplication: (AB)C = A(BC) (associative law) (A + B)C = AC + BC (distributive law #1) C(A + B) = CA + CB (distributive law #2) (ra)b = A(rB) = r(ab) Any of the above identities holds provided that matrix sums and products are well defined.

13 If A and B are n n matrices, then both AB and BA are well defined n n matrices. However, in general, AB BA. ( ) ( ) Example. Let A =, B = ( ) ( ) Then AB =, BA = If AB does equal BA, we say that the matrices A and B commute.

14 Problem. Let A and B be arbitrary n n matrices. Is it true that (A B)(A + B) = A 2 B 2? (A B)(A + B) = (A B)A + (A B)B = (AA BA) + (AB BB) = A 2 + AB BA B 2 Hence (A B)(A + B) = A 2 B 2 if and only if A commutes with B.

15 Diagonal matrices If A = (a ij ) is a square matrix, then the entries a ii are called diagonal entries. A square matrix is called diagonal if all non-diagonal entries are zeros Example , denoted diag(7, 1, 2) Let A = diag(s 1, s 2,...,s n ), B = diag(t 1, t 2,...,t n ). Then A + B = diag(s 1 + t 1, s 2 + t 2,...,s n + t n ), ra = diag(rs 1, rs 2,...,rs n ).

16 Example = Theorem Let A = diag(s 1, s 2,...,s n ), B = diag(t 1, t 2,...,t n ). Then A + B = diag(s 1 + t 1, s 2 + t 2,...,s n + t n ), ra = diag(rs 1, rs 2,...,rs n ). AB = diag(s 1 t 1, s 2 t 2,...,s n t n ). In particular, diagonal matrices always commute (i.e., AB = BA).

17 Example a 11 a 12 a 13 7a 11 7a 12 7a a 21 a 22 a 23 = a 21 a 22 a a 31 a 32 a 33 2a 31 2a 32 2a 33 Theorem Let D = diag(d 1, d 2,...,d m ) and A be an m n matrix. Then the matrix DA is obtained from A by multiplying the ith row by d i for i = 1, 2,...,m: v 1 v 2 A =. v m d 1 v 1 d 2 v 2 = DA =. d m v m

18 Example. a 11 a 12 a a 11 a 12 2a 13 a 21 a 22 a = 7a 21 a 22 2a 23 a 31 a 32 a a 31 a 32 2a 33 Theorem Let D = diag(d 1, d 2,...,d n ) and A be an m n matrix. Then the matrix AD is obtained from A by multiplying the ith column by d i for i = 1, 2,...,n: A = (w 1,w 2,...,w n ) = AD = (d 1 w 1, d 2 w 2,...,d n w n )

19 Identity matrix Definition. The identity matrix (or unit matrix) is a diagonal matrix with all diagonal entries equal to 1. The n n identity matrix is denoted I n or simply I. I 1 = (1), I 2 = ( ) 1 0, I = In general, I = Theorem. Let A be an arbitrary m n matrix. Then I m A = AI n = A.

20 Inverse matrix Let M n (R) denote the set of all n n matrices with real entries. We can add, subtract, and multiply elements of M n (R). What about division? Definition. Let A be an n n matrix. Suppose there exists an n n matrix B such that AB = BA = I n. Then the matrix A is called invertible and B is called the inverse of A (denoted A 1 ). A non-invertible square matrix is called singular. AA 1 = A 1 A = I

21 Examples ( ) 1 1 A =, B = 0 1 ( ) 1 1, C = 0 1 ( ) ( ) ( ) AB = =, ( ) ( ) ( ) BA = =, ( ) ( ) ( ) C 2 = = ( ) Thus A 1 = B, B 1 = A, and C 1 = C.

22 Inverting diagonal matrices Theorem A diagonal matrix D = diag(d 1,...,d n ) is invertible if and only if all diagonal entries are nonzero: d i 0 for 1 i n. If D is invertible then D 1 = diag(d1 1 1,...,dn ). d d d n 1 = d dn 1 d 1

23 Inverting 2 2 matrices Definition. ( ) The determinant of a 2 2 matrix a b A = is det A = ad bc. c d ( ) a b Theorem A matrix A = is invertible if c d and only if det A 0. If det A 0 then ( ) 1 a b = c d 1 ad bc ( ) d b. c a

24 System of n linear equations in n variables: a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 Ax = b, a n1 x 1 + a n2 x a nn x n = b n where a 11 a a 1n x 1 b 1 a 21 a a 2n A =......, x = x 2., b = b 2.. a n1 a n2... a nn Theorem If the matrix A is invertible then the system has a unique solution, which is x = A 1 b. x n b n