# Why do we need invariants to represent real life quantities?

Often it is said that one of the most useful properties of eigenvalues of a matrix is that they are invariant under change of basis. This in turn is said to be useful in physics because real, physical quantities represented by the matrix/tensor do not care about the coordinate system used to measure them. Therefore what we measure in real life needs to be independent of the coordinate system. Components of a tensor/matrix change with a change of basis, so we need to use the invariants of the tensor/matrix if the want to get correct results. This is what I think is true.

I'm having a hard time understanding this. I'm studying structural mechanics at the moment, where we write stresses in form of a tensor. The 3x3 stress tensor represents stresses at each point inside an object with normal stresses in diagonal and shear stresses off diagonal. Each column of the stress tensor gives a vector of stresses for each side of an infinitesimal "box" element. But the components of this tensor change with a change in coordinate system. Why is this a problem?

Let's say a particle has the velocity vector $$(1, 1)$$ in regular orthogonal cartesian coordinates. If we introduce another rectangular coordinate system that is rotated 90 degrees counter-clockwise, the particle now has the velocity vector $$(1, -1)$$. Even though the vector has different components, neither vector is wrong: They both represent the same velocity on real life.

So why we need to resort to using invariants of a tensor to calculate things in real life if the components of the tensor do represent the same thing, but just using different coordinate system? Thank you!

EDIT:

The components $${\displaystyle \sigma _{ij}}$$ of the stress tensor depend on the orientation of the coordinate system at the point under consideration. However, the stress tensor itself is a physical quantity and as such, it is independent of the coordinate system chosen to represent it.

and:

The value of these components will depend on the coordinate system chosen to represent the vector, but the magnitude of the vector is a physical quantity (a scalar) and is independent of the Cartesian coordinate system chosen to represent the vector.

In addition, I got an answer on my earlier question on Engineering SE stating that independence from coordinate system is necessary, or any conclusions drawn from the analysis are wrong.