What is the connection between inertia tensor and rotational inertia?

[ehrenfest]

Homework Statement

What is the relationship between the inertia tensor and the rotational inertia? How can a second-rank tensor describe the same thing as a scalar? Is their a formula to go between one and the other i.e. if I have an inertia tensor for a rigid body rotating about some axis, can I get the rational inertia of the body about that axis from that?

EDIT: I see. Please notice that the inertia tensor IS DEFINED WITH RESPECT TO A COORDINATE SYSTEM while a moment of inertia IS DEFINED WITH RESPECT TO AN AXIS. In fact, the inertia tensor contains the information about the rotational inertias about the 3 axes of the coordinate with respect to which it is defined. These are simply the diagonals. Please confirm this.
EDIT EDIT: If I have an inertia tensor with respect to a coordinate system with axes $$\hat{i},\hat{j},\hat{k}$$, how do you get the moment of inertia about an axis that is not equal to one of $$\hat{i},\hat{j},\hat{k}$$?

Homework Equations

The Attempt at a Solution

 

[Mindscrape]

Are you asking about is how to diagonalize the inertia tensor to use principle axes or asking about Steiner's parallel axis theorem?