7.3.3 Tensor Component Index Address

Tensor Component Index Address is the specific tuple of index values that identifies the location of a single component within a tensor's array, serving the same role for a tensor that a coordinate pair serves for a point on a grid. It is the addressing scheme by which any one entry of a tensor's component structure can be located, retrieved, or referenced without ambiguity.

Definition and Scope

Tuples as Addresses

For a tensor of type ((p,q)) on an (n)-dimensional space, an index address is an ordered tuple

with each entry ranging over (1,\dots,n), and the semicolon separating upper from lower indices to preserve the distinction between contravariant and covariant slots.

Uniqueness of Address to Value

Each valid index address corresponds to exactly one scalar entry in the component array; no two distinct addresses point to the same storage location unless a symmetry has been imposed that forces their values to coincide, in which case the addresses remain distinct even though the stored values agree.

Structural Properties

Ordering and Position Within the Index Slots

The order of the indices in the address is significant whenever the corresponding slots are not exchangeable. For a tensor lacking symmetry between its first and second lower indices, the addresses ((i; 1, 2)) and ((i; 2, 1)) refer to two different scalar entries; only when a stated symmetry links those slots do the two addresses become linked.

Mapping to Linear Storage

In computational representations, a multi-index address is typically converted to a single linear memory offset through a fixed formula, commonly a row-major or column-major scheme. For a rank-2 tensor of dimension (n) stored row-major, the address ((i,j)) maps to the offset

generalizing directly to higher rank by nesting the same construction one index at a time.

Partial Addresses

An index address need not fix every slot. Leaving one or more indices unspecified turns the address into a description of a component area, a sub-block of entries sharing the fixed part of the address while varying over the unfixed part, extending the addressing scheme from single entries to ranges of entries.

Role Within Tensor Algebra

Alignment in Operations

Tensor operations are defined by how they act on index addresses. Contraction identifies pairs of addresses that share a value in one upper and one lower slot and sums over that shared value; the tensor product concatenates the address of one tensor with the address of another to form the address of the result.

Practical Use in Implementation

Index addressing is the mechanism by which software libraries expose tensor indexing syntax to users, translating a request such as accessing the entry at a given tuple into the correct memory offset under whatever storage layout the library uses internally, while shielding the caller from the details of how that address is resolved.