that is the dot product of the row vector A and a row vector B is equal to
the matrix product of A and the column vector BT. This observation
helps to prove our properties.
1. Notice that the dot product <A,B>=ABT is a scalar, that is a 1 by 1 matrix. Thus <A,B> is a symmetric matrix (every 1 by 1 matrix is obviously symmetric). So
Here we used several times the theorem about
transposes. Of course, this statement could be easily proved without
the theorem about transposes (just use the definition of the dot product
of n-vectors) but I wanted to demonstrate the relationship between
the dot product and the matrix product.
2. Again we use the properties of the matrix product:
We leave property 3 as an exercise.
In order to prove property 4, we need to use the definition of the dot product: Let A=(a1,...,an). Then