12.7.3 Tensor Equality Component Criterion

Tensor Equality Component Criterion is the practical test stating that two tensors of the same type, expressed relative to the same basis, are equal precisely when every scalar component of the first tensor matches the corresponding scalar component of the second tensor at each identical index position.

Statement of the Criterion

Componentwise Matching Condition

For tensors $$A$$ and $$B$$ of type $$(p,q)$$, expressed in a common basis, the criterion states:

for every possible assignment of index values across all upper and lower positions. Every single component, without exception, must match for equality to hold.

Exhaustiveness of the Check

The criterion requires agreement at every one of the components, not merely at a subset. A tensor with even one mismatched component, however small the discrepancy, fails to satisfy the criterion and is therefore not equal to the tensor being compared against.

Application to Simple Cases

Vectors

For vectors treated as rank-1 tensors, the component criterion reduces to checking that every entry matches:

for each index $$i$$, which is precisely the ordinary notion of two vectors being equal when their coordinates coincide.

Rank-2 Tensors

For rank-2 tensors, the criterion requires that every entry of the corresponding component array match at the same row and column:

for all valid values of $$i$$ and $$j$$.

Relationship to Other Aspects of Equality Verification

Prerequisite Conditions

The component criterion presumes that the same type requirement and the same space requirement are already satisfied, since a componentwise comparison is only meaningful once both tensors share an identical index structure and originate from the same underlying vector space.

Basis Independent Guarantee

Once the component criterion is verified in one basis, the result extends to every other basis automatically, since the transformation law linking different bases is linear and therefore preserves equality once established.

Practical Use as a Computational Test

Direct Numerical Comparison

In practice, the component criterion offers a straightforward, purely numerical way to test equality: list out the components of both tensors in a shared basis and check each pair for an exact match, without needing any further geometric or algebraic reasoning.

Equivalent Zero-Difference Test

An equivalent formulation of the criterion checks whether every component of the difference tensor $$A-B$$ equals zero, which is logically identical to checking that every component of $$A$$ matches the corresponding component of $$B$$.

Illustration