Multiplying a matrix by its transpose

[g.lemaitre]

Homework Statement

I don't see how you multiply a matrix by its transpose. If a matrix is 3 x 2 then its transpose is 2 x 3. I thought you couldn't multiply matrices unless they have the same rows and columns.

 

[Saitama]

For matrices to be multiplied, the condition is that the number of columns of the first matrix should be equal to the number of rows of the other matrix.

 

[genericusrnme]

Nope, if a matrix A is n x m and B is m x l then AB is defined

If you do the same procedure of matric multiplication you'll see that multiplying a 3 x 2 and a 2 x 3 matrix gives you a 3 x 3 matrix of rank at most 2
If you multiply 2 x 3 by 3 x 2 you'll get a 2 x 2 matrix with rank, also (and obviously), at most 2

In terms of components if A = BC, where B is n x m and C is m x l, then

$A_{i,k} = \sum_{j=1}^m B_{i,j} C_{j,k}$