12.20.2 Tensor Scalar Multiplication Notation

Tensor scalar multiplication notation defines how scalars interact with tensors, using indices and symbols to represent scaling operations in multilinear algebra.

Tensor Scalar Multiplication Notation is the symbolic convention used to represent the operation of multiplying a tensor by a scalar, producing a new tensor of the same order and shape whose components are each scaled by the given scalar value. This notation preserves the index structure of the original tensor while indicating that every component undergoes the same uniform scaling factor.

Definition and Basic Form

Symbolic Representation

Tensor scalar multiplication is typically written by placing the scalar directly adjacent to the tensor symbol, with no explicit operator between them, following the same convention used in vector and matrix scalar multiplication.

B = λ A

Here, $λ$ denotes the scalar, $A$ denotes the original tensor, and $B$ denotes the resulting tensor after scaling.

Index Notation Form

When tensors are expressed using indicial notation, the scalar multiplies each component individually while all upper and lower indices remain unchanged in position and count.

B_{j}^{i} = λ A_{j}^{i}

This expresses that the component of $B$ with contravariant index $i$ and covariant index $j$ equals the scalar multiplied by the corresponding component of $A$ .

Structural Properties Reflected in Notation

Order Preservation

The notation does not introduce or remove any indices, which reflects the property that scalar multiplication never changes the order (rank) of a tensor. A tensor of order two remains of order two after multiplication by a scalar.

Componentwise Application

The scalar is understood to apply to every component of the tensor simultaneously, rather than to a subset of components. Notationally, this is implied by writing the scalar once outside the tensor symbol rather than attaching it to individual index positions.

for all valid i, j : B_{j}^{i} = λ A_{j}^{i}

Notational Variants

Explicit Multiplication Symbol

In some texts, particularly introductory treatments, an explicit multiplication symbol is retained to emphasize the operation before the reader becomes accustomed to the implicit juxtaposition convention.

B = λ \cdot A

Placement of the Scalar

The scalar may be written before or after the tensor symbol without changing meaning, since scalar multiplication commutes with the tensor.

λ A = A λ

Distinction from Related Notations

Difference from Tensor Product Notation

Scalar multiplication notation must not be confused with the tensor product symbol, since the tensor product combines two tensors into a higher-order tensor, whereas scalar multiplication combines a scalar and a tensor into a tensor of the same order.

Difference from Contraction Notation

Scalar multiplication notation involves no repeated indices and therefore invokes no summation convention, distinguishing it clearly from contraction notation, where repeated upper and lower indices imply summation over that index.

Consistency with Linear Structure

The notation for tensor scalar multiplication is consistent with the broader linear-algebraic notation used for vector spaces, since tensors of a fixed order and type form a vector space under addition and scalar multiplication. This consistency allows scalar multiplication notation for tensors to directly generalize the familiar notation used for scalars multiplying vectors and matrices.