14.9.4 Tensor Kronecker Component Layout
The Tensor Kronecker Component Layout arranges tensor components in a matrix format, enabling efficient computation and clarity in multilinear algebra operations.
Tensor Kronecker Component Layout is the description of how individual scalar components of two factor tensors are physically arranged into the single array of components that represents their Kronecker product. It specifies which component of the first factor scales which block of the second factor, and how those scaled blocks are stitched together into rows and columns of a single composite array.
Component Placement Rule
Scaling of Blocks by Outer Entries
Each entry of the first factor multiplies an entire copy of the second factor, and the resulting scaled copy is placed as one contiguous block within the composite layout. The position of that block within the overall array mirrors the position of the scaling entry within the first factor.
Location Within the Composite Array
The row and column position of any component in the composite array is obtained by combining the outer position from the first factor with the inner position from the second factor, using the extents of the second factor as the step size between successive outer positions.
Layout Diagram
Grid of Scaled Blocks
The diagram below shows how a two-by-two first factor produces a grid of four scaled copies of the second factor, each occupying its own contiguous region of the composite layout.
Row-Major and Column-Major Layout
Row-Major Component Sequencing
Under a row-major layout, components are laid out block-row by block-row, with each block-row containing the full horizontal span of scaled blocks corresponding to a single row of the first factor before advancing to the next block-row.
Column-Major Component Sequencing
Under a column-major layout, the same components are laid out block-column by block-column instead, with the internal arrangement of each block also reversed to column-major order, so that the overall linear storage sequence differs from the row-major case while the logical array remains identical.
Internal Layout Within a Block
Preservation of the Second Factor's Structure
Within any single block, the relative arrangement of components is an exact copy of the internal layout of the second factor, scaled uniformly by the single outer entry associated with that block. No component from one block influences the values placed inside a different block.
Boundary Alignment Between Adjacent Blocks
Adjacent blocks in the composite layout share no components; the last column of components in one block-column is immediately followed by the first column of components in the next block-column, with no gap or overlap between them.
Layout Consistency Under Transposition
Transposing the Composite Array
Transposing the composite layout as a whole produces the same result as transposing each factor individually and then forming the Kronecker product of the transposed factors, which shows that the block structure of the layout is compatible with the transpose operation.
Effect on Block Positions
Transposition swaps the row and column position of every block as a whole, and simultaneously transposes the internal layout of the contents within each block, so both the outer grid and the inner arrangement are affected consistently.
Layout Under Repeated Kronecker Products
Nested Block Structure
When a Kronecker product involves three or more factors, the component layout becomes a nested structure: each top-level block, determined by the first factor, itself subdivides into blocks determined by the second factor, and those subdivide further according to the third factor, continuing recursively for additional factors.
Recovering Individual Factor Components
Given the position of a single component within the fully nested layout, the corresponding component of every individual factor can be recovered by successively extracting the outer and inner indices at each level of nesting, following the same step-size rule used for two factors.