Vector Multiplication

The multiplication of a vector by a vector produces some interesting results, known as the vector inner product and as the vector outer product.

Prerequisite: This material assumes familiarity with matrix multiplication.

Vector Inner Product

Assume that a and b are vectors, each with the same number of elements. Then, the inner product of a and b is s.

a'b = b'a = s

where
      a and b are column vectors, each having n elements,
      a' is the transpose of a, which makes a' a row vector,
      b' is the transpose of b, which makes b' a row vector, and
      s is a scalar; that is, s is a real number - not a matrix.

Note this interesting result. The product of two matrices is usually another matrix. However, the inner product of two vectors is different. It results in a real number - not a matrix. This is illustrated below.

Then,

a'b = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32

Thus, the inner product of a'b is equal to 32.

Note: The inner product is also known as the dot product or as the scalar product.

Vector Outer Product

Assume that a and b are vectors. Then, the outer product of a and b is C.

ab'= C

where
      a is a column vector, having m elements,
      b is a column vector, having n elements,
      b' is the transpose of b, which makes b' a row vector, and
      C is a rectangular m x n matrix

Unlike the inner product, the outer product of two vectors produces a rectangular matrix, not a scalar. This is illustrated below.

Then,

Notice that the elements of Matrix C consist of the product of elements from Vector A crossed with elements from Vector B. Thus, Matrix C winds up being a matrix of cross products from the two vectors.