How this instrument works
The Hadamard product pairs A and B position by position: entry (1,1) of A meets only entry (1,1) of B, entry (1,2) meets only entry (1,2), and so on. There is no summing across a row or down a column — the operation never looks past the single matching cell. That is the entire definition, and it is why the two matrices must share an identical shape: with no row-to-column pairing to reconcile, a 2×2 simply has nothing to multiply against a 2×3.
Ordinary matrix multiplication treats a matrix as a linear map — a machine for turning one vector into another — so it combines a whole row of A against a whole column of B. The Hadamard product ignores that machinery entirely and treats A and B as flat grids of numbers, the same way a spreadsheet multiplies two ranges of cells or a programming array multiplies two lists term by term. Take A = [[1,2],[3,4]] and B = [[5,6],[7,8]]: the Hadamard product gives [[5,12],[21,32]], while the ordinary matrix product of the same pair gives [[19,22],[43,50]] — same two matrices, two entirely different, equally valid, answers.
One consequence surprises people who expect elementwise operations to be the 'simple' cousin: the Schur product theorem, proved by Issai Schur, states that if A and B are both positive semi-definite, their Hadamard product is guaranteed positive semi-definite too — a structural fact about the linear-map world surviving an operation that supposedly ignores it. At the plain edges, the all-ones matrix acts as the identity (A ⊙ ones leaves A unchanged) and the zero matrix is absorbing (A ⊙ 0 is 0 everywhere), which is exactly why this product doubles as a masking tool in image and neural-network work.
- Enter the first matrix's four numbers into A: row1,col1, A: row1,col2, A: row2,col1, and A: row2,col2.
- Enter the second matrix, matching shape, into B: row1,col1, B: row1,col2, B: row2,col1, and B: row2,col2.
- Read the four Product fields — Product: row1,col1 is simply A: row1,col1 times B: row1,col1, and each other cell follows the same same-position rule.
- Change one entry and watch only its matching Product cell move; the other three hold still, a quick check that no row-column mixing is happening.
Worked example — [[1,2],[3,4]] against [[5,6],[7,8]]
Set A: row1,col1 = 1, A: row1,col2 = 2, A: row2,col1 = 3, A: row2,col2 = 4, and B: row1,col1 = 5, B: row1,col2 = 6, B: row2,col1 = 7, B: row2,col2 = 8. Each Product cell is its own pair multiplied and nothing else: Product: row1,col1 = 1 × 5 = 5, Product: row1,col2 = 2 × 6 = 12, Product: row2,col1 = 3 × 7 = 21, and Product: row2,col2 = 4 × 8 = 32.
The full result reads P = [[5,12],[21,32]]. It is worth naming what this is not: the ordinary matrix product of the same A and B works out to [[19,22],[43,50]], built from row-times-column sums instead of matching cells. Two legitimate operations, two different tables of numbers, from the exact same starting matrices — which is the whole reason this instrument exists as its own page rather than a mode on a general matrix multiplier.
Questions
How is the Hadamard product different from regular matrix multiplication?
Regular matrix multiplication sums row-times-column products, so (AB) at position (1,1) mixes every entry of A's first row with every entry of B's first column. The Hadamard product only ever multiplies the two entries sitting in the identical position — for A = [[1,2],[3,4]] and B = [[5,6],[7,8]] that gives [[5,12],[21,32]], while ordinary multiplication of the same pair gives [[19,22],[43,50]].
Why is it called the Hadamard product?
It carries the name of Jacques Hadamard, the French mathematician who studied it, and is also called the Schur product after Issai Schur, who proved that the Hadamard product of two positive semi-definite matrices stays positive semi-definite. Both names refer to the same element-wise operation; you will see either in a textbook.
What matrix sizes does the Hadamard product accept?
Both matrices must be exactly the same shape — same row count and same column count — because every entry needs a partner sitting in the identical position. This differs from ordinary matrix multiplication, which only needs the inner dimensions to match and happily accepts, say, a 2×3 against a 3×4. A 2×2 Hadamard-multiplied against a 2×3 is simply undefined.
What is the identity element for the Hadamard product?
A matrix filled entirely with 1's, not the usual identity matrix with 1's only on the diagonal — Hadamard-multiplying any matrix by the all-ones matrix returns it unchanged, since each entry is multiplied by 1. The zero matrix behaves the opposite way, as the absorbing element: multiplying by it zeroes every entry out, which is exactly how a binary mask switches values off.
Where does the Hadamard product actually get used?
Neural networks lean on it constantly — an LSTM or GRU gate multiplies a signal vector against a same-shaped gate vector of values between 0 and 1, entry by entry, to decide how much of each signal passes through. Image processing uses it to apply a pixel mask, and any 'scale this array by that array, position for position' computation is a Hadamard product by definition.
Does the Hadamard product commute like ordinary multiplication of numbers?
Yes — A ⊙ B always equals B ⊙ A, because multiplying two numbers in a given position never depends on which one is written first. It is also associative and distributes over addition, so (A ⊙ B) ⊙ C = A ⊙ (B ⊙ C) and A ⊙ (B + C) = (A ⊙ B) + (A ⊙ C). Ordinary matrix multiplication has none of those guarantees for order.