Mathematics - The MathWorks - #20

Multiplying Matrices

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 singleasterisktodenotematrixmultiplication. Thenexttwo examples illustrate the fact that matrix multiplication is not commutative; AB is usually not equal to BA:

X = A*B

X=

Y =

Y =

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

u= [3; 1;4];