Tensor Product Preview

Math Linear Algebra • Applications and Advanced Linear Algebra (capstone)

Written by STEM Calculators Team

Compute the tensor product \(A \otimes B\). For vectors, this is the outer product matrix \(C_{ij}=a_i b_j\). For matrices, this is the Kronecker product (a matrix representation of the tensor product of linear maps), with size \((mp)\times(nq)\) when \(A\in\mathbb{R}^{m\times n}\) and \(B\in\mathbb{R}^{p\times q}\). Includes a contraction / multi-linear identity demo.

Mode

A type

B type

Dimensions

A rows / len

B rows / len

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Tolerance

Display decimals

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Inputs accept -3.5, 2e-4, fractions 7/3, and constants pi, e.

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Results

Result shape

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Summary

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Contraction check

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Note

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Result

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Key formulas

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Tensor grid visual

Heatmap of the computed matrix result. Zoom with wheel/trackpad, drag to pan, double-click to reset. Play animates reveal.

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Choose inputs and click “Calculate”.

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Theory

Tensor product vs. outer product vs. Kronecker product

The tensor product is a construction that combines vector spaces into a larger “product space” that can represent multilinear behavior. At a concrete computational level, you often see two closely related objects:

For vectors \(a\in\mathbb{R}^m\) and \(b\in\mathbb{R}^p\), the tensor product (in this preview) is the outer product matrix \(C=a\otimes b\in\mathbb{R}^{m\times p}\) defined by \[ C_{ij}=a_i b_j. \] This matrix has rank 1 whenever \(a\neq 0\) and \(b\neq 0\).
For matrices \(A\in\mathbb{R}^{m\times n}\) and \(B\in\mathbb{R}^{p\times q}\), a standard matrix representation of the tensor product of linear maps is the Kronecker product \(K=A\otimes B\in\mathbb{R}^{(mp)\times(nq)}\). It is built as a block matrix: each entry \(A_{ij}\) becomes a block \(A_{ij}B\).

In many applied areas, the Kronecker product is exactly what people mean by “tensor product of matrices,” because it behaves correctly with respect to vectorization, basis changes, and multilinear identities.

Entry formula and block structure

If \(A\in\mathbb{R}^{m\times n}\) and \(B\in\mathbb{R}^{p\times q}\), the Kronecker product \(K=A\otimes B\) satisfies the index rule (using 1-based indices): \[ K_{(i-1)p+r,\,(j-1)q+s} = A_{ij}\,B_{rs}. \] This can be read as: pick an entry of \(A\), multiply the entire matrix \(B\) by it, and place that block into \(K\). The output size expands from \(m\times n\) and \(p\times q\) to \((mp)\times(nq)\), so the operation increases the “rank/size” of the representation.

For vectors \(a\in\mathbb{R}^m\), \(b\in\mathbb{R}^p\), the outer product \(a\otimes b\) is a special case: it produces a matrix with columns proportional to \(a\) (scaled by components of \(b\)).

Contraction (reducing order)

While tensor products increase order, contraction reduces order by summing over a matched index. In the simplest setting, if \(C=a\otimes b\) is an outer product matrix and \(w\in\mathbb{R}^p\), then \[ Cw = (a\otimes b)w = a(b\cdot w). \] The scalar \(b\cdot w\) collapses the \(b\)-index, leaving a vector in the \(a\)-space. This is exactly the idea behind “mode contraction” used throughout multilinear algebra and tensor methods.

Key multilinear identity (why Kronecker is useful)

One of the most important identities is how Kronecker products act on Kronecker-structured vectors. If \(A\in\mathbb{R}^{m\times n}\), \(B\in\mathbb{R}^{p\times q}\), \(x\in\mathbb{R}^{n}\), and \(y\in\mathbb{R}^{q}\), then \[ (A\otimes B)(x\otimes y) = (Ax)\otimes(By). \] This is a clean “multilinear map” statement: the big matrix \(A\otimes B\) acts on the big vector \(x\otimes y\) in a way that separates into two smaller matrix-vector products.

This idea appears in numerical PDEs, tensor networks, and quantum computing. In quantum mechanics, the tensor product combines the state spaces of subsystems; tensor-product structure is also where “entanglement” comes from. In machine learning, Kronecker/tensor products can represent feature interactions and structured linear layers.

Frequently Asked Questions

Is the Kronecker product the same as the tensor product?

For matrices viewed as linear maps, the Kronecker product is the standard matrix representation of their tensor product in chosen bases.

Why does the output size grow so fast?

Because indices multiply: (m×n) with (p×q) becomes (mp)×(nq). Tensor products encode joint degrees of freedom.

What does contraction mean here?

Contraction sums over one index (like a dot product along a mode). It reduces the order/dimension of a tensor expression.

Where is this used in practice?

Kronecker/tensor products appear in multilinear algebra, tensor networks, quantum computing (composite systems/entanglement), and structured ML models.

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