12.7.5 Tensor Equality Evaluation Criterion
Understanding when two tensors are equal, based on their components and the underlying space they reside in.
Tensor Equality Evaluation Criterion is the overall standard combining the type requirement, the space requirement, and the component matching condition into a single ordered procedure used to determine, in practice, whether two given tensors should be treated as the same object.
The Complete Evaluation Procedure
Step One: Verify Matching Type
The first stage of evaluation checks whether the two tensors and share the same type , meaning the same number of contravariant and covariant indices with matching dimension throughout. If this fails, evaluation terminates immediately with the conclusion that the tensors are unequal, since no further comparison is meaningful.
Step Two: Verify Same Underlying Space
Once type is confirmed to match, evaluation proceeds to check that both tensors are built over the same underlying vector space and its corresponding dual space. If the tensors originate from unrelated vector spaces, the comparison is considered undefined rather than a straightforward inequality, since no canonical correspondence between their respective bases exists.
Step Three: Componentwise Comparison in a Shared Basis
With type and space confirmed to match, the tensors are expressed relative to a single common basis, and their corresponding scalar components are compared at every index position:
Equality is confirmed only if every such comparison succeeds; a single mismatch at any index position is sufficient to conclude that the tensors are unequal.
Why an Ordered Procedure Is Useful
Avoiding Wasted Computation
Checking type first is computationally cheap and immediately rules out the vast majority of unequal tensor pairs before any detailed componentwise work is needed. Only tensors that pass this initial filter warrant the more involved space and component checks.
Preventing Ill-Defined Comparisons
Checking the underlying space before attempting a componentwise comparison prevents the error of comparing numbers that have no canonical correspondence to one another, which could otherwise produce a spurious answer to a question that is not actually well posed.
Guarantee Provided by the Full Criterion
Consistency with Basis Independence
Because the componentwise stage is carried out in a single shared basis, and the underlying transformation law linking bases is linear, satisfying this evaluation criterion in one basis guarantees that the same conclusion, whether equal or unequal, holds in every other basis as well.
Alignment with the Reflexive, Symmetric, and Transitive Properties
The evaluation criterion, when satisfied, defines a genuine equivalence relation among tensors of the same type and space: every tensor is equal to itself, equality is symmetric between any two tensors, and equality is transitive across any chain of tensors, all of which follow directly from the underlying equality of numbers used in the componentwise stage.