1.  = 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

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

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

4.  = 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)

5.  = 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)

6.  = 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)

7.  = 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)

8.  = 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

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

10. 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