What is the meaning of $\oplus$ and $\otimes$?

I am struggling to fully understand some notation in a book where they use a "crosshair" symbol - first like $\bigoplus\limits_{i=1}^n{} Z_j$ where the $Z_j$ are matrices and second like $I_n \otimes \Phi$ where $I_n$ and $\Phi$ are both matrices.

The book is about multivariate statistics and the section is about random coefficient models. There isn't a notation/terminology appendix to refer to. I was going to post a digital pic of the page so that users can see the context (this is at the start of the section).

So is this on topic here or should I post on math.se ?

Update: I originally posted this on meta.se and it was migrated here. I am now attaching the photo from the relevant page of the book.

• $\oplus$, Direct sum; $\otimes$, Tensor product.
– whuber
Commented Oct 14, 2014 at 15:41
• @whuber thanks ! That's exactly what I needed. Did you want to make it into an answer so I can accept it ? Commented Oct 14, 2014 at 15:53
• I'm too busy now, Joe--a real answer would explain what these things are, rather than resorting to links. If anyone would care to provide the details in an answer I'd be happy to vote it up, but in the meantime I am glad you can move on with your reading.
– whuber
Commented Oct 14, 2014 at 15:54
• @whuber OK I will upload the digital picture I mentioned to give more context Commented Oct 14, 2014 at 16:14

In statistics, $$A\oplus B:=\left[\begin{array}{cc}A & 0 \\ 0 & B \end{array} \right]$$ and (e.g. for a $2\times 2$-matrix $A$) $$A\otimes B:=\left[\begin{array}{cc}a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \end{array} \right].$$

This focuses on matrices for their use in statistics as design or hypothesis matrices etc., where these notations simplify the frequent block structure of such matrices. One can find the name Kronecker sum for $\oplus$ and Kronecker product for $\otimes$, especially in the manuals of statistical software. (Also very handy is the component wise matrix multiplication $A\#B=[a_{ij}b_{ij}]_{i,j}$ for equally shaped matrices. It's sometimes called Hadamard product.)

In mathematics, $\oplus$ and $\otimes$ have their slightly different typical meaning as direct sum or tensor product of vector spaces or even more general algebraic structures.

• The statistical meaning is identical to the mathematical meaning of direct sum and tensor product of linear transformations: these formulas result from writing out the transformations as matrices in a particular basis.
– whuber
Commented Oct 15, 2014 at 18:46
• The theorem holds that the set of $n\times m$-matrices is isomorphic to the set of linear transformations from a $m$ dimensional to a $n$ dimensional vector space. So in the vector space world you are right. But a direct sum exists also between structures like groups. In statistics, the latter is usually not meant. Commented Oct 16, 2014 at 8:18
• @whuber when I first searched for this I came upon [this] (math.stackexchange.com/questions/207635/…) which confused me quite a bit and prompted me to ask the question on CV. Is that link actually showing the same usage, because the accepted anser there says it is "non carrying addition" ? Commented Oct 16, 2014 at 8:19
• @HorstGrünbusch I just wanted to alert you to the comment I just made above toe whuber - also, if you could update your answer to reflect what the meanings are called in statistics I would be happy to accept it. Commented Oct 16, 2014 at 8:23
• We should be glad that mathematical ideas are far richer than our ability to make orthographic distinctions. Few mathematical symbols--and none of the simple one-character ones--have universal meanings. You have to understand them in context. Although $\oplus$ is universally understood as direct sum in a linear algebraic context, computer scientists use it to distinguish addition of bit strings-cum-vectors over $\mathbb F_2$ from addition of bit strings that stand for integers (i.e., without and with carries). The $\wedge$ ("wedge", not "caret"!) similarly has many meanings.
– whuber
Commented Oct 16, 2014 at 14:23

Suppose $$A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$$ and $$B=\begin{pmatrix}e&f\\g&h\end{pmatrix}$$

Then,

A$$\otimes$$B=$$\begin{pmatrix}a&b\\c&d\end{pmatrix}$$ $$\otimes$$ $$\begin{pmatrix}e&f\\g&h\end{pmatrix}$$=$$\begin{pmatrix} a {\begin{pmatrix}e&f\\g&h\end{pmatrix}}&b{\begin{pmatrix}e&f\\g&h\end{pmatrix}}\\c{\begin{pmatrix}e&f\\g&h\end{pmatrix}}&d{\begin{pmatrix}e&f\\g&h\end{pmatrix}}\end{pmatrix}$$

And

A$$\oplus B=\begin{bmatrix}A&0\\0&B\end{bmatrix}$$

• I have just answered this question to make it more clear and understandable.. It's my first answer on this platform Commented May 26, 2023 at 13:59
• Oh cool, welcome to the site! I just asked because I don't see the added value: your answer is basically the same answer to what was there before Commented May 26, 2023 at 14:18
• Thanks, I will try to add more value to this if I found something important Commented May 26, 2023 at 14:31