Coordinate transformation matrix?

[Will_C]

Can anyone tell me:
1) How to understand the defination to orthogonal transformation matrix?
Defination: A(i,j)A(k,j)=q(i,k) where q is Kronecker delta.
2) Why the inverse of this orthogonal matrix is equal to its transpose?

Will.

 

[matt grime]

In short if we want to find the pq'th entry of a product of matrices then

$$(AB)_{p,q}= \sum_r A_{p,r}B_{r,q}:=A_{p,r}B_{r,q}$$

the convention being that when ever we see a repeated index we sum.

What does A(i,j)A(k,j) mean? well the (k,j)th entry of A is the jk'th entry in A transpose, so what you've written is the same as (AA^t)(i,k) and states that

"the ik'th entry of AA^t is 1 if i=k, and zero otherwise"#

which is exactly what it means to be the identity matrix.

Thus 1 and 2 are exactly the same thing.

 

[Will_C]

Excuse me, matt grime,
I am not quite understand what you mentioned above.
Would you mind make it simply or explain it more?
BTW, I don't know how to input math symbol (such as summation sign, subscript...) in the thread.

Thx,
Will.

 

[matt grime]

Let's try and see where the problem is:

Have you met the notation that

$$A_{i,j}$$

is the entry in row i column j of a matrix?

cick on the maths to see how to typeset it.

Did you try and work through some small examples, such as 2x2 matrices to see how this notation does indeed show how they multiply together?

If A and B are 2x2 matrices then, as we all know,

$$(AB)_{1,1} = A_{1,1}B_{1,1} + A_{1,2}B_{2,1}$$

which is exactly what I wrote with the summation sign. You've done summation signs right?


Then entry in row i column j of A^t is the same as the entry in row j column i of A.

Do you see that?