Mathematics

Matrix Multiplication

Multiplication of matrices is defined in a way that reflects composition of the underlying linear transformations and allows compact representation of systems of simultaneous linear equations. The matrix product C = AB is defined when the column dimension of A is equal to the row dimension of B, or when one of them is a scalar. If A is m-by-p and B is p-by-n, their product C is m-by-n. The product can actually be defined using MATLAB for loops, colon notation, and vector dot products.

• A = pascal(3);
  B = magic(3);
  m = 3; n = 3;
  for i = 1:m
       for j = 1:n
          C(i,j) = A(i,:)*B(:,j);
       end
  end 
  

MATLAB uses a single asterisk to denote matrix multiplication. The next two examples illustrate the fact that matrix multiplication is not commutative; AB is usually not equal to BA.

• X = A*B
  
  X =
        15    15    15
        26    38    26
        41    70    39
  
  Y = B*A
  
  Y =
        15    28    47
        15    34    60
        15    28    43
  

A matrix can be multiplied on the right by a column vector and on the left by a row vector.

• u = [3; 1; 4];
  x = A*u
  
  x =
         8
        17
        30
  
  v = [2 0 -1];
  y = v*B
  
  y =
        12    -7    10
  

Rectangular matrix multiplications must satisfy the dimension compatibility conditions.

• C = fix(10*rand(3,2));
  X = A*C
  
  X =
        17    19
        31    41
        51    70
  
  Y = C*A
  
  Error using ==> *
  Inner matrix dimensions must agree.
  

Anything can be multiplied by a scalar.

• s = 7;
  w = s*v
  
  w =
        14     0    -7
  

Vector Productsand Transpose

The Identity Matrix