How Do Direction Cosines Relate to the Kronecker Delta in Matrix Multiplication?

[sameershah23]

I have to prove the following.
lmj. lsj= delta ms m,j,s=1,2,3.
and lmj and lsj stands for direction cosine matrix. and (delta ms) is a Kronecker delta.


2. Cant think of any

The Attempt at a Solution

 

[D H]

Your nomenclature is very confusing. Is this one transformation matrix L indexed by m and j on one hand and indexed by s and j on the other? What is that dot? I have a guess regarding what you are trying to prove here, but it is just a guess. Please clarify.

 

[sameershah23]

l`s denote direction cosines and the dot represents the multiplication of the two matrices. And m,s,j take the values 1,2,3.
I guess you know the Kronecker delta means,

 

[D H]

You have not clarified thing much. This nomenclature problem, too me, reflects a lack of understanding. You are not talking about "the multiplication of two matrices" because the product of two random direction cosine matrices is not the identity matrix. The standard matrix product of a direction cosine matrix with itself is not the identity matrix. The matrix product of a direction cosine matrix and one other matrix is the identity matrix. Why all this talk about identity matrices? Because \delta_{i,j} is just another way to write the identity matrix.

Some hints:

• Each column (and each row, for that matter) of a direction cosine matrix represents something very important. What does a column in a direction cosine matrix represent?
• What is the dot product of \hat {\boldsymbol i} with itself? With {\boldsymbol j} or {\boldsymbol k}?
• How do the above two questions relate to the problem at hand?

It is getting late. Could someone else take over helping this person?