# What does this symbol $\otimes$ mean?

I came across the symbol $\otimes$ as below and I would like to know what this symbol $\otimes$ means:

$\text{.... the projection operator P is given by: }$ $$P = I_nd - \nabla G^T(\nabla G \nabla G^T)^{-1} \nabla G= I_{nd} - I_d \otimes uu^T,$$ where $I_x$ denotes the identity matrix of size $x\times x$ and $\mathbf{u}$ is the unit vector , $\mathbf{u} = (1,1,1,\dots,1)/\sqrt{n})$ in $\mathbb{R}^n$

## 4 Answers

In this context, $\otimes$ refers specifically to the Kronecker product. In particular, we have $$I_d \otimes B = \overbrace{B \oplus B \oplus \dots \oplus B}^d = \text{diag}(\overbrace{B,B, \dots, B}^d)\\ = \pmatrix{B\\&B\\&&\ddots\\&&&B}$$

Here is an official reference, more or less, from Dummit and Foote's Abstract Algebra (3rd. ed., p. 421):

Let $A=(\alpha_{ij})$ and $B$ be $r\times n$ and $s\times m$ matrices, respectively, with coefficients from any commutative ring. The Kronecker product or tensor product of $A$ and $B$, denoted by $A\otimes B$, is the $rs\times nm$ matrix consisting of an $r\times n$ block matrix whose $i, j$ block is the $s\times m$ matrix $\alpha_{ij} B$.

It means the tensor product...

Tensor Product. Check here for reference. http://en.wikipedia.org/wiki/Tensor_product

EDIT: Ninja'd