How this instrument works
Given two vectors a = (a1, a2) and b = (b1, b2), the tensor product — also called the outer product — multiplies every component of a against every component of b and keeps each of the four results as its own separate entry: t1 = a1×b1, t2 = a1×b2, t3 = a2×b1, and t4 = a2×b2, arranged as a 2×2 matrix rather than added together. Contrast that with the dot product, which multiplies matching components and adds the two results into one number; the tensor product performs the multiplying but skips the adding entirely, so the output has more entries than either input rather than fewer.
This site's Convolution page looks superficially similar but does something genuinely different, and the contrast is worth spelling out precisely. Convolving two 2-term sequences produces three outputs — c0 = a1b1, c1 = a1b2 + a2b1, c2 = a2b2 — because its middle term deliberately sums two of the four raw products, a1b2 and a2b1, into one combined figure. The tensor product never performs that addition: those same two quantities stay apart forever, reported here as t2 and t3 rather than folded into a single c1. One operation collapses four products down to three numbers by adding a matched pair together; the other keeps all four products intact and simply names each one.
Because nothing gets summed, the tensor product raises rather than lowers dimension — two flat, 2-component vectors become one 2×2 matrix, a genuinely higher-dimensional object built from every combination of their entries. This structure is the starting block behind rank-one matrices in linear algebra, behind combining independent quantum states in physics, and behind building larger feature grids from two smaller ones in numerical work — anywhere the full grid of pairwise combinations matters more than any single combined total.
A useful pattern shows up whenever a vector is combined with itself: multiplying (2, 3) against (2, 3) gives 4, 6, 6, and 9, and the two middle entries always match in that case, since a1×a2 equals a2×a1 either way round. The resulting matrix, [[4, 6], [6, 9]], is symmetric across its diagonal — a direct, visible consequence of ordinary multiplication not caring which factor comes first.
- Enter the first vector's two numbers into Vector a: first component and Vector a: second component.
- Enter the second vector's two numbers into Vector b: first component and Vector b: second component.
- Read Product: a₁b₁ and Product: a₁b₂ for the top row of the resulting matrix.
- Read Product: a₂b₁ and Product: a₂b₂ for the bottom row.
- Set either vector's second component to zero to watch an entire row of the output collapse to zero as well.
Worked example — (1, 2) tensor (3, 4)
Set Vector a to (1, 2) and Vector b to (3, 4). Every pairing gets multiplied and kept on its own: t1 = 1×3 = 3, t2 = 1×4 = 4, t3 = 2×3 = 6, and t4 = 2×4 = 8, giving the matrix [[3, 4], [6, 8]]. Feed that identical pair of vectors into this site's Convolution sheet instead and its middle term reads 10 — exactly t2 + t3, or 4 + 6 — because convolution adds those two cross terms together where this page reports them as two separate numbers, 4 and 6, never combined.
A second case shows what happens when a component is zero: (1, 0) tensor (5, 6) gives 1×5 = 5, 1×6 = 6, 0×5 = 0, and 0×6 = 0, so the matrix [[5, 6], [0, 0]] has an entire bottom row of zeros. Nothing about the top row changes; only the row driven by the zero component of vector a vanishes, since any number multiplied by 0 stays at 0.
Questions
What is a tensor product of two vectors?
Multiplying every component of one vector against every component of another and keeping all four results separate, arranged as a 2×2 matrix rather than combined into one figure. For (a1, a2) and (b1, b2) that's a1b1, a1b2, a2b1, and a2b2, each one its own matrix entry.
How is the tensor product different from the dot product?
The dot product multiplies matching components and adds the results into a single number; the tensor product multiplies every component of one vector against every component of the other and keeps all four results separate. One shrinks two vectors down to a scalar, the other expands them into a matrix.
How is this different from the site's convolution calculator?
Convolution takes the same four raw products this page computes and sums two of them together — its middle output, c1 = a1b2 + a2b1, folds two of those four figures into one number. This page performs no such addition: every one of the four stays reported on its own, so a 2×2 matrix comes out instead of a 3-term sequence.
What is (1, 2) tensor (3, 4)?
The matrix [[3, 4], [6, 8]]. Each entry is a single product — 1×3 = 3, 1×4 = 4, 2×3 = 6, and 2×4 = 8 — with no addition combining any of the four figures.
Why does (1, 0) tensor (5, 6) produce a row of zeros?
Because the second component of vector a is 0, and any number multiplied by 0 is 0 — that zero passes straight through to 0×5 = 0 and 0×6 = 0, wiping out the entire bottom row while leaving the top row, driven by the nonzero first component, untouched.
Is the tensor product of two vectors symmetric?
Only when the two vectors are identical. Tensoring a vector with itself, such as (2, 3) with (2, 3), gives a matrix where the two middle entries match — 6 and 6 here — because a1×a2 and a2×a1 are the same value. With two different vectors the middle entries generally differ.