Tensor Product Calculator — Outer Product of Two Vectors
Calculate the tensor product (outer product) of two 2-dimensional vectors u and v. The result is a 2×2 matrix where entry (i, j) = uᵢ × vⱼ. Enter the components of both vectors and get the full matrix, Frobenius norm, trace (equal to the dot product), and determinant — with a step-by-step solution.
Equals ‖u‖ × ‖v‖ — the product of the two vector magnitudes
Entry at row 1, column 1
Entry at row 1, column 2
Entry at row 2, column 1
Entry at row 2, column 2
Result matrix u ⊗ v
- 1
‖u‖ = √(u₁² + u₂²)
√(3² + 1²) = 3.1623 - 2
‖v‖ = √(v₁² + v₂²)
√(2² + 4²) = 4.4721 - 3
Frobenius norm = ‖u‖ × ‖v‖
3.1623 × 4.4721 = 14.1421For any rank-1 matrix (outer product), the Frobenius norm equals the product of the vector magnitudes.
How does this calculator work?
The outer (tensor) product of u = [u₁, u₂] and v = [v₁, v₂] is a 2×2 matrix with Mᵢⱼ = uᵢvⱼ: result = [[u₁v₁, u₁v₂], [u₂v₁, u₂v₂]]. The matrix has rank 1, determinant 0, trace equal to the dot product u·v, and Frobenius norm equal to ‖u‖×‖v‖.
Formula
How this is calculated
The outer product (tensor product) of two column vectors u ∈ ℝᵐ and v ∈ ℝⁿ produces an m×n matrix M where Mᵢⱼ = uᵢ vⱼ. For 2-D vectors this gives a 2×2 matrix. Unlike the dot product (which collapses two vectors into a scalar), the outer product encodes how each component of u scales each component of v.
The resulting matrix always has rank at most 1 — its columns are all scalar multiples of u, and its rows are all scalar multiples of v. This means its determinant is always zero. A key identity: the trace of u ⊗ v equals the dot product u · v. The Frobenius norm ‖u ⊗ v‖_F = ‖u‖ × ‖v‖, which can be verified by squaring and summing all four entries.
Outer products are fundamental in linear algebra: they appear in the definition of the tensor product of vector spaces, in the construction of projection matrices (P = uuᵀ / uᵀu), in Singular Value Decomposition (SVD) where a matrix decomposes as a sum of outer products), and in neural network weight updates (outer product of gradient vectors). This calculator demonstrates the 2-D case; the principle extends identically to higher dimensions.
Frequently asked questions
The dot product u · v multiplies corresponding components and sums the results, yielding a single scalar that measures how aligned the vectors are. The outer product u ⊗ v multiplies every component of u with every component of v, yielding a matrix that captures the full pairwise interaction.
The outer product u ⊗ v has rank 1 — all its columns are multiples of u and all its rows are multiples of v, so its column space is one-dimensional. A matrix with rank less than its order has a zero determinant. Geometrically, the transformation collapses all of ℝ² onto a line (the span of u).
Projection matrices use outer products: P = uuᵀ / (uᵀu) projects any vector onto the line spanned by u. SVD decomposes a matrix as M = Σᵢ σᵢ (uᵢ ⊗ vᵢ), a sum of rank-1 outer products. In machine learning, weight gradient updates in fully-connected layers are computed as an outer product of the error signal and the input activation.
Also known as
TG we-Calculate Editorial Team. (2026). Tensor Product Calculator — Outer Product of Two Vectors [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/tensor-product-calculator
TG we-Calculate Editorial Team. "Tensor Product Calculator — Outer Product of Two Vectors." TG we-Calculate. 2026. https://we-calculate.com/calculator/tensor-product-calculator.
TG we-Calculate Editorial Team, "Tensor Product Calculator — Outer Product of Two Vectors," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/tensor-product-calculator
@misc{wecalculate_tensor_product_calculator, title = {Tensor Product Calculator — Outer Product of Two Vectors}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/tensor-product-calculator}}, year = {2026}, note = {TG we-Calculate} }
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