Understanding Tensor Calculations: Exploring Equations 4.74 and 4.76

[Kiwi1]

Two questions have me a bit stumped.

I am given:
eqn 4.74 $J^{i}_{i'} J^{i'}_{j} =\delta^{i}_{j}$ and
eqn 4.76 $J^{i'}_{i} J^{i}_{j'} =\delta^{i'}_{j'}$

Problem 47:
Derive equation 4.76 from 4.74 by multiplying both sides by $J^j_{j'}$. I have gone in a bunch of circles but can't get there.Problem 62:
Explain why the covariant basis, interpreted as a matrix is positive definite.

I don't see how I can express the covariant basis as a matrix without first defining an ambient cartesian coordinate syste. A no-no. I wonder if the question is in error? The problem is in a section about the Covariant Metric Tensor and that can easily be expressed as a square matrix and shown to be positive definite.

 

[jvicens]

I can definitely understand why these questions have you stumped. I will do my best to provide some guidance and explanation for both problems.

Problem 47:
To derive equation 4.76 from 4.74, we need to manipulate the given equations by using the properties of matrices. First, let's rewrite equation 4.74 as:

$J^{i}_{i'} J^{i'}_{j} =\delta^{i}_{j}$

Now, we can multiply both sides by $J^j_{j'}$:

$J^{i}_{i'} J^{i'}_{j} J^j_{j'} =\delta^{i}_{j} J^j_{j'}$

Using the associative property of matrix multiplication, we can rearrange the left side as:

$J^{i}_{i'} \left(J^{i'}_{j} J^j_{j'}\right) =\delta^{i}_{j} J^j_{j'}$

Now, we can substitute equation 4.76 into this expression:

$J^{i}_{i'} \left(\delta^{i'}_{j'}\right) =\delta^{i}_{j} J^j_{j'}$

Simplifying, we get:

$J^{i}_{j'} =\delta^{i}_{j} J^j_{j'}$

And finally, using the definition of the Kronecker delta, we arrive at equation 4.76:

$J^{i'}_{i} J^{i}_{j'} =\delta^{i'}_{j'}$

I hope this helps you understand the derivation process. It's important to remember that when working with matrices, we can use various properties to manipulate the equations and arrive at the desired result.

Problem 62:
The covariant basis can be interpreted as a matrix by expressing it in terms of the basis vectors in a given coordinate system. For example, in a Cartesian coordinate system, the covariant basis can be written as a matrix with the basis vectors as its columns. This matrix will be a square matrix, and each of its diagonal elements will be the squared magnitude of the corresponding basis vector.

To understand why this matrix is positive definite, we need to look at its eigenvalues. A positive definite matrix is defined as a matrix whose eigenvalues are all positive. In this case, the eigenvalues of the covariant basis matrix will be the squared