Perform tensor products, contractions, and index manipulations on rank-2 and rank-3 tensors with visual component grids.
Fill in the components of a rank-2 or rank-3 tensor as grids of numbers, choose an operation, and the tool returns the result as a new component grid. Tensor products pair every component of one grid with every component of the other and raise the rank; contractions sum over a chosen index pair and lower it. The surprise is that the ordinary matrix product is itself a contraction - a sum over the shared index - so 'product' here does not multiply components one by one. Everything is computed in the browser from the grids you enter.
A concrete input and expected output from the current implementation.
Input
A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]] as rank-2 grids; operation: contract the shared index (matrix product A_ij B_jk)
Expected output
A 2x2 grid with rows [19, 22] and [43, 50].
Contraction sums over the repeated index j: (AB)_ik = A_i1 B_1k + A_i2 B_2k. Entry (AB)_11 = 1x5 + 2x7 = 19; the other entries are 1x6 + 2x8 = 22, 3x5 + 4x7 = 43, and 3x6 + 4x8 = 50. Two free indices remain, so the output keeps the same 2x2 shape.
A rank-2 tensor is entered as a two-dimensional grid and a rank-3 tensor as a stack of component grids, with rows and columns standing for the first and second index. The tool carries that layout through the computation, so the returned grid has one axis per free (unsummed) index. That is why identical numbers can give different results when entered transposed.
A tensor product multiplies every component of one grid by every component of the other, so two 2x2 grids combine into 16 components. A contraction instead selects one index on each factor (or two indices of one tensor), sums over them, and drops the summed axes. Contracting the shared index of two 2x2 grids is exactly matrix multiplication; contracting a single 2x2 grid over both indices is its trace.
No. Mathematically, the tensor product of two rank-2 tensors has rank 4 and 16 components for 2x2 inputs, while the matrix product contracts the shared index to a rank-2 grid, (AB)_ik = A_ij B_jk. In this tool they are separate operations with different output sizes, so choose by what rank you want back.
A contraction sums over one pair of indices, so a single contraction of a rank-3 tensor leaves one free index: the result is a rank-1 tensor, shown as a one-dimensional grid. There are three pairs of indices you can sum, and they generally give three different vectors. Contracting twice would need two index pairs, which a rank-3 tensor does not have.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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