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's for loops, colon notation, and vector dot products.

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.

x = A*u

x =
       8
      17
      30

y = v*B

y =
      12    -7    10

Rectangular matrix multiplications must satisfy the dimension compatibility conditions.

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.

w = s*v

w =
      14     0    -7

Vector Products and Transpose

The Identity Matrix