12.8 Tensor Operation Composition Procedure

Tensor Operation Composition Procedure is the systematic method by which several individual tensor operations, such as addition, subtraction, scalar multiplication, and negation, are combined in sequence to build more complex tensor expressions, following a well-defined order that respects the type and space compatibility requirements of each intermediate step.

Structuring a Composed Operation

Sequential Application of Simple Operations

A composed expression involving multiple tensors, such as $$kA+lB-C$$, is evaluated by applying the individual operations in a definite sequence: first the scalar multiplications producing $$kA$$ and $$lB$$, then the negation producing $$-C$$, and finally the additions combining all three resulting tensors into a single sum.

Maintaining Type Consistency at Every Step

At each stage of the composition, the intermediate result must remain a tensor of a definite, well-defined type. Since scalar multiplication and negation both preserve type, and addition requires operands of matching type, every intermediate tensor produced during the composition must be checked to ensure it shares the type of whatever tensor it will next be combined with.

Order of Operations

Precedence Among Tensor Operations

Scalar multiplication and negation, being unary or single-tensor operations, are generally applied before addition or subtraction, mirroring the usual precedence conventions from ordinary algebra. This ensures that expressions such as $$k(A+B)$$ and $$kA+kB$$ are handled consistently, since the distributive law guarantees these two orders of composition yield the same result.

Grouping and Parentheses

When an expression contains parentheses, the operations within the innermost parentheses are composed first, following exactly the same principle used in ordinary arithmetic, ensuring the composition procedure produces an unambiguous final tensor regardless of how the expression is written.

Verifying Compatibility Throughout Composition

Checking Same Type at Each Addition or Subtraction

Every time an addition or subtraction is performed as part of a larger composed expression, the same type requirement and the same space requirement must be verified for the two tensors being combined at that step, not merely for the overall final result.

Propagating Type Through Scalar Operations

Because scalar multiplication and negation always preserve the type of their operand, tracking the type of a tensor through a chain of such operations is straightforward: the type entering a scalar multiplication or negation step is identical to the type leaving it, simplifying the overall bookkeeping needed during composition.

Basis Independence of the Composed Result

Consistency Across Every Step

Since each individual operation used in the composition, whether addition, subtraction, scalar multiplication, or negation, commutes with the linear transformation law for change of basis, the entire composed procedure also commutes with change of basis. The final tensor produced by the composition, when expressed in any basis, matches the result obtained by transforming each individual operation's output into that same basis.

Guarantee of a Well-Defined Final Tensor

This consistency ensures that a composed tensor expression represents one single, unambiguous abstract tensor, regardless of which basis was used during the intermediate steps of the computation.

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