# Origin of the symbol for the tensor product

I have recently realised that the Paleo-Hebrew (and Phoenician) graph for the Hebrew letter ט (Teth) is $$\otimes$$. This made me wonder if there is any relation between the choice of the symbol and the choice of the name (as far as I am aware, tensor product is meant to be a stretched product, coming from latin tendere although I confess I can't understand why the tensor product of two spaces, or modules, or elements, should be thought as being "stretched"), since ט is the letter used in modern and ancient Hebrew to denote the sound T in words borrowed from other languages.

I have checked the Archives Bourbaki, in particular the second redaction (n°034) of Algèbre. Chapitre II, algèbre linéaire and the corresponding discussion which seem to me the first occurrences of the name (still called "produit tensoriel (ou kroneckerien)", see ibid. p. 198) and of the symbol, but found no hints. The tensor product is presented as a special case of the bilinear product of two modules, which is denoted by $$\odot$$, itself not a graph that I am aware of in the Paleo-Hebrew alphabet: the circle in $$\otimes$$ could simply come from the one in $$\odot$$, but it might also be related to the first letter T of tensor.

• on the origin of the word "tensor" in mathematics see: jeff560.tripod.com/t.html – Michael Bächtold Dec 2 '20 at 11:43
• The symbol $\oplus$ was used in Linear B to write the syllable ka- in Mycenean Greek. Like the above, this would seem to be only a coincidence. – Robert Furber Dec 2 '20 at 14:11
• It should be noted that throughout history, nearly every culture that has ever been has had some kind of circle as its first symbol, some kind of cross as its second symbol, and some combination of the two as a very early development thereafter. There are easily hundreds of independent symbols that can be generally described as a an X inscribed in a circle. – KRyan Dec 2 '20 at 18:09
• @KRyan To me, my X-Men ... – Yemon Choi Dec 2 '20 at 22:20
• @PaulTaylor It might very well be that the symbol of $\Theta$ has some common origin with the Paleo-Hebrew symbol for $\otimes$: after all, the Greek alphabet is derived from the Phoenician, which is the same alphabet used in Paleo-Hebrew, and Theta is the same letter as Theth. Interesting observation, thanks! That being said, you're the first bringing religion into the picture. – Filippo Alberto Edoardo Dec 16 '20 at 8:07

According to John Aldrich's list of "Earliest Uses of Symbols for Matrices and Vectors", the notation $$\times$$ for direct product (as well as the name itself) goes back to Wedderburn's 1934 Lectures on Matrices (page 74).

Some further search gave a much earlier source, Hurwitz's 1894 Zur Invariantentheorie

I still have to track down the step from $$\times$$ to $$\otimes$$.
Incidentally, On the history of the Kronecker product argues that it should more appropriately be called the Zehfuss product.

In any case, since tensor product is $$\otimes$$ and tensor sum is $$\oplus$$, it seems obvious that the $$\times$$ and the $$+$$ refer to the arithmetical operations of multiplication and addition, not to a letter.

• This is not a direct product. – darij grinberg Dec 2 '20 at 12:24
• please correct me if I'm wrong, but I'm following this terminology --- which I understand is the one in the OP: Kronecker product = matrix direct product = $\otimes$ – Carlo Beenakker Dec 2 '20 at 12:30
• Huh, never heard of this terminology. It does not correspond to the direct product of vector spaces. – darij grinberg Dec 2 '20 at 12:38
• But it does correspond to the tensor product $S \otimes T : W \otimes X \to Y \otimes Z$ of linear maps $S : W \to Y$ and $T : X \to Z$. – Branimir Ćaćić Dec 2 '20 at 12:43
• It is my understanding that historically "direct product" has sometimes been used to mean what is now known as "tensor product". Many physicists still conflate the two terminologies, e.g. see physics.stackexchange.com/questions/447342/… – Carmeister Dec 2 '20 at 20:03