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Advanced Matrix Operations and Array Addressing Techniques

Explore sophisticated matrix operations, array addressing methods, and vector functions to enhance your data manipulation skills in numerical computing. Learn about matrix multiplication, identity matrix, inverse matrix, and element-by-element calculations. Discover practical examples and solve complex equations efficiently.

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Advanced Matrix Operations and Array Addressing Techniques

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  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. TransposeInterchange 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 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. 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 Inverse of a Matrix

  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. 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 Division Example

  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)

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