1. Week 3 Last week Vectors Array Addressing Mathematical Operations Array Multiplication and Division Identity Matrix Inverse of a Matrix Element by Element Calculations

2. Array Addressing A = 1 2 3 4 5 6 7 8 9 >> B = A([2 3],[2 3]) B = 5 6 8 9 >> B = A(2:3,2:3) B = 5 6 8 9

3. Vectors >> q = 0:0.1:2*pi; % semicolon suppresses output >> y=sin(q); % Comment after % is ignored >> plot(q,y)

4. Transpose Interchange rows and columns >> X = [1 2 3 4 5 6 7 8 9 10] X = 1 2 3 4 5 6 7 8 9 10 >> X' ans = 1 6 2 7 3 8 4 9 5 10

5. Vector Functions A = [ 1 2 3 4 5 6] sum(A) = 21 max(A) = 6 min(A) = 1

6. Random Arrays >> A = rand(4,3) A = 0.9501 0.8913 0.8214 0.2311 0.7621 0.4447 0.6068 0.4565 0.6154 0.4860 0.0185 0.7919

7. Sound Files sound_in.m creates an array called data. data is an array of one column with 10,000 rows Sounds can be saved under different names Yes = data; and = data; No = data; Arrays can be concatenated to form longer sounds yesandNo = [Yes; and; No]

8. Array Multiplication Y = A*B The number of columns in A must equal the number of rows in B.

9. Multiplication Example C(1,1) = 1*1 + 2*2 + 3*3 = 14 C(1,2) = 1*2 + 2*1 + 3*1 = 7 C(2,1) = 3*1 + 2*2 + 1*3 = 10 C(2,2) = 3*2 + 2*1 + 1*1 = 9 C(3,1) = C(3,2) = C(4,1) = C(4,2) =

10. Errors A = 1 2 3 4 5 6 7 8 9 >> C= ones(3,2) C = 1 1 1 1 >> A*C ans = 6 6 15 15 24 24 >> C*A ??? Error using ==> * Inner matrix dimensions must agree.

11. Example >> BV = [3; 1; 4] BV = 3 1 4 >> AV = [2 5 1] AV = 2 5 1 % AV*BV is the dot product >> AV*BV% The number of columns in ans =% AV equals the number of 15% rows in BV % AV*BV is a scalar >> BV*AV% Rows in BV = 3 ans =% Columns in AV = 3 6 15 3% BV&AV is a 3x3 array 2 5 1 8 20 4

12. Identity Matrix >> I = eye(3) I = 1 0 0 0 1 0 0 0 1

13. Identity Matrix Multiplying a matrix by the identity matrix is equivalent to multiplication by one >> I = eye(3) I = 1 0 0 0 1 0 0 0 1 If A is square AI = IA = A

14. Inverse of a Matrix The matrix B is the inverse of the matrix A if BA = AB = I where I is the identity matrix >> A =[ 1 2; 2 3] A = 1 2 2 3 >> B = A^-1 B = -3 2 2 -1 >> A*B ans = 1 0 0 1

15. Array Division Solve for X where A*X = B A and B are known arrays. A^-1 *A*X = A^-1 *B Since A^-1* A = I X = A^-1* B Left Division X = A\B

16. Division Example 2x - 3y + 4z = 10 x + 6y - 3z = 4 -5x + y + 2z = 3 A*X = B where A = 2 -3 4 1 6 -3 -5 1 2 B = 10 4 3 >> X = A\B X = 1.2609 2.2261 3.5391 >> 2*X(1) -3*X(2) + 4*X(3) ans = 10

17. Element by Element Operations If the usual symbols (* / ^) are used operations follow the rules of linear algebra. Sometimes we don’t want this. Given x = [1 2 3] And y = [2 4 6] Find z = [x(1)*y(1), x(2)*y(2), x(3)*y(3)] Use the dot product operator.* z = x.*y = [2 8 18]

18. Element by Element Operators Multiplication.* x(n) = A(n)*B(n) Division./x(n) = A(n)/B(n) Exponentiation.^x(n) = A(n)^y(n) Left Division.\ x(n) = B(n)/A(n) x = A.*B x = A./ B x = A.^ y x = A.\B

19. Element by Element Example A = [ 0 1 2 3 4] B = [ 5 4 3 2 1] >> A.*B ans = 0 4 6 6 4 >> A./B ans = 0 0.2500 0.6667 1.5000 4.0000 >> A.^2 ans = 0 1 4 9 16

20. Element by Element Calculation >> x= 0:0.1:10; >> y = x.^2; >> plot(x,y)