It is true for the anti-symmetric part also because the sign change cancels.hellomrrobot said:I am having trouble showing that ##(A_{ik}B_{kj})_{mm} = (A_{ki}B_{jk})_{mm}##Wouldn't the right side end up having a different outcome? Or can we assume its symmetric?
Mentz114 said:It is true for the anti-symmetric part also because the sign change cancels.
Yes, I think so. A and B can be decomposed into symmetric and anti-symmetric parts.hellomrrobot said:anti-symmetric?
It would help if you would explain your notation.hellomrrobot said:I am having trouble showing that ##(A_{ik}B_{kj})_{mm} = (A_{ki}B_{jk})_{mm}##Wouldn't the right side end up having a different outcome? Or can we assume its symmetric?
gill1109 said:It would help if you would explain your notation.
I suppose that you are using Einstein summation convention. And that when you write (A_{ik}B_{kj}) you mean, the matrix whose i,j element is sum_k (A_{ik}B_{kj}). In other words, the ij element of the matrix AB. Now you want to sum over m the m,m elements of that matrix, so you are talking about trace(AB).
Similarly on the right hand side, replace i and j both by m and m, sum over m and sum over k, and you see that what you have written is just trace(AB).
Thank you! You are right.stevendaryl said:Well, I would say that the right hand side is trace(BA), but that is always equal to trace(AB).
Index notation is a mathematical notation used to represent and manipulate quantities with multiple indices. It is commonly used in physics and engineering to simplify and generalize complex equations. In proving equations, index notation allows for a concise and elegant representation of the relationship between different terms.
2.The subscript "mm" in this equation represents the specific element in the mth row and mth column of the matrix. In index notation, the indices indicate the position or location of the terms within a matrix. Therefore, (A_{ik}B_{kj})_{mm} refers to the element in the mth row and mth column of the matrix resulting from the multiplication of A and B.
3.The order of indices is different in these two expressions because of the commutative property of matrix multiplication. In other words, the order of multiplication does not affect the result. Therefore, (A_{ik}B_{kj})_{mm} and (A_{ki}B_{jk})_{mm} are equivalent expressions, but with different index notation representations.
4.Yes, this equation can be extended to higher dimensions. In fact, index notation is particularly useful when working with higher dimensional matrices, as it allows for a more compact and efficient representation of equations. The same principles of index notation can be applied to prove equations involving matrices with more than two dimensions.
5.This equation can be applied in various real-world situations, particularly in physics and engineering. For example, it can be used to prove certain relationships between physical quantities, such as force and acceleration, or to solve problems involving matrix operations in engineering applications. Index notation is a powerful tool for representing and manipulating complex equations, making it a valuable tool for scientists and researchers in various fields.