 = N  N matrix multiplication N = 3 matrix N = 3 matrix N = 3 matrix

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Presentation transcript:

 = N  N matrix multiplication N = 3 matrix N = 3 matrix N = 3 matrix Matrix C a11 a12 a13 b11 b12 b13  = a21 a22 a23 b21 b22 b23 a31 a32 a33 b31 b32 b33 Matrix A Matrix B 1 12 8 -1 23 4

 = N  N matrix multiplication N = 3 matrix Matrix A Matrix B Matrix C C11 = (a11  b11) + (a12  b21) + (a13  b31)

 = N  N matrix multiplication N = 3 matrix Matrix A Matrix B Matrix C C11 = (a11  b11) + (a12  b21) + (a13  b31)

 = N  N matrix multiplication N = 3 matrix Matrix A Matrix B Matrix C C11 = (a11  b11) + (a21  b21) + (a13  b31) C21 = (a21  b11) + (a22  b21) + (a23  b31)

 = N = 3 matrix Matrix A Matrix B Matrix C a11 a21 a31 a12 a22 a32 (a11  b11) + (a21  b21) + (a13  b31) C21 = (a21  b11) + (a22  b21) + (a23  b31) C31 = (a31  b11) + (a32  b21) + (a33  b31)

 = N = 3 matrix Matrix A Matrix B Matrix C a11 a21 a31 a12 a22 a32 (a11  b11) + (a21  b21) + (a13  b31) C12 = (a11  b12) + (a12  b22) + (a13  b32) C21 = (a21  b11) + (a22  b21) + (a23  b31) C31 = (a31  b11) + (a32  b21) + (a33  b31)

 = N = 3 matrix Matrix A Matrix B Matrix C a11 a21 a31 a12 a22 a32 (a11  b11) + (a21  b21) + (a13  b31) C12 = (a11  b12) + (a12  b22) + (a13  b32) C21 = (a21  b11) + (a22  b21) + (a23  b31) C22 = (a21  b12) + (a22  b22) + (a23  b32) C31 = (a31  b11) + (a32  b21) + (a33  b31)

 = How many additions and multiplications are needed for completing a N  N matrix multiplication? a11 a21 a31 a12 a22 a32 a13 a23 a33 N = 3 matrix b11 b21 b31 b12 b22 b32 b13 b23 b33 Matrix A Matrix B  = c11 c21 c31 c12 c22 c32 c13 c23 c33 Matrix C These imply that a N  N matrix multiplication is a N3 algorithm (i.e., O(N3) algorithm) C11 = (a11  b11) + (a21  b12) + (a13  b31) C12 = (a11  b12) + (a12  b22) + (a13  b32) C21 = (a21  b11) + (a22  b21) + (a23  b31) C22 = (a21  b12) + (a22  b22) + (a23  b32) C31 = (a31  b11) + (a32  b21) + (a33  b31) (a) N2  N = N3 multiplications C33 = (a31  b13) + (a32  b23) + (a33  b33) (b) N2  (N-1) = N3 – N2 additions

For N = 4 C11 = (a11 × b11) + (a21 × b12) + (a31 × b13) + (a41 × b14) + + C11

Applications and algorithms that use N  N matrix multiplications w/ huge data sets  Discrete Fourier transform (DFT) - for spectral analysis Spectral analysis is the technical process of decomposing a complex signal into simpler parts.  Computer graphics - for image processing  Quantum mechanics  Graph theory  Solving linear equations