Are These Vector and Matrix Operations Formulated Correctly?

[oxxiissiixxo]

w=∇×u Is this correct? w_i=ε_ijk ∂/(∂x_j ) u_k
w and u are the vectors

C=(x∙y)z Is this correct? C_i= ∑_i〖(x_i y_j)∙z_i 〗
C, x, y, z are vectors

A^T∙A ∙x=A^T∙b Is this correct? A_ij^T∙A_ij∙x_j=A_ij^T∙b_i
A is tensor and x and b are vectors

A^T is A transpose

 

[HallsofIvy]

Assuming you are not using subscripts and superscripts to distinguish between a vector and its transpose, yes, that is correct.

 

[Phrak]

C=(x∙y)z Is this correct? C_i= ∑_i〖(x_i y_j)∙z_i 〗
C, x, y, z are vectors

Hall is usually right on, though this one should be (assuming orthonormal coordinates)
C=(x∙y)z C_j = z_i ∑_j (x_j y_j)

 

[oxxiissiixxo]

I want to make sure the way I am writing this A^T dot A dot x = A^T dot b in index notation correctly.

Would you mind to do that one time for me so that I can match up with my answer?

My answer was

A^T dot A dot x = A^T dot b >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

>>>>>>>>>>>>>>> A_ij^T dot A_ij dot x_j = A_ij^T dot b_i

A_ij^T mean A_ij is a tensor with ij components and A^T mean the transpose of A
x and b are both vector

This problem was asking me to rewrite the A^T dot A dot x = A^T dot b in index notation. Thank you!

 

[Phrak]

I want to make sure the way I am writing this A^T dot A dot x = A^T dot b in index notation correctly.

Would you mind to do that one time for me so that I can match up with my answer?

My answer was

A^T dot A dot x = A^T dot b >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

>>>>>>>>>>>>>>> A_ij^T dot A_ij dot x_j = A_ij^T dot b_i

A_ij^T mean A_ij is a tensor with ij components and A^T mean the transpose of A
x and b are both vector

This problem was asking me to rewrite the A^T dot A dot x = A^T dot b in index notation. Thank you!

Just a quick hint before I knock off. The transpose of a matrix interchanges rows with columns. (A_ij)^T = A_ji.

To make the conversion to index notation it's convenient to think of the first index as indexing rows and the second as indexing columns:

C dot D = ∑_j (C_ij D_jk)

 

[oxxiissiixxo]

Is this correct? ∑_ j A_ji dot A_ij dot x_j = ∑_ i A_ij^T dot b_i ?

 

[Phrak]

The summation should occur between only two tensors. As you've written it, you are summing over all three on the left hand side.

I should have noticed this earlier, but you should place the dot operator only between two vectors.

U·V = UV^T = ∑_i U_i V_i

Vector U is multiplied with the transpose of vector V.

So we should write the expression A^T·Ax as

A^TAx^T, where A is a matrix, A^TA is a matrix, and x is a row vector.

A^TA is equal to A_ij^T multiplied by A_ij, which means we are going to multiply the rows of the first matrix by the rows of the second

A^TA = ∑_i A_ij A_ij

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