Tensor Product Properties

Question

I am currently working my way through some notes and have got stuck proving a couple of tensor product properties. I have A,B,C,D as matrices and u,x,y as vectors, with a & b being constants.

I have managed to prove most of the results that arise using vectors:

(x+y)$\otimes$u = x$\otimes$u + y$\otimes$u,

u$\otimes$(x+y) = u$\otimes$x + u$\otimes$y, &

ax$\otimes$by = ab(x$\otimes$y)

However, I am struggling to prove the following properties:

(A$\otimes$B)(C$\otimes$D) = AC$\otimes$BD,

(A$\otimes$B)(x$\otimes$y) = Ax$\otimes$By, &

(A$\otimes$B)$^\dagger$ = A$^\dagger$$\otimes$B$^\dagger$

Answer 1

For the first one, simply go and verify that \begin{align} (A \otimes B)(C \otimes D) &= \begin{bmatrix} a_{11}B &\ldots&a_{1n}B \\ \vdots & \ddots & \vdots\\ a_{m1}B & \ldots & a_{mn}B \\ \end{bmatrix} \begin{bmatrix} c_{11}D &\ldots&c_{1p}D \\ \vdots & \ddots & \vdots\\ c_{n1}D & \ldots & c_{np}D \\ \end{bmatrix}\\ &= \begin{bmatrix} \sum_{k=1}^na_{1k}c_{k1}BD &\sum_{k=1}^na_{1k}c_{kp}BD \\ \vdots & \ddots & \vdots\\ \sum_{k=1}^na_{mk}c_{k1}BD & \sum_{k=1}^na_{mk}c_{kp}BD \\ \end{bmatrix} \end{align} And the result then follows.

Answer 2

For (A$\otimes$B)(x$\otimes$y), using the same approach as above:

\begin{align} (A \otimes B)(x \otimes y) &= \begin{bmatrix} a_{11}B &\ldots&a_{1n}B \\ \vdots & \ddots & \vdots\\ a_{m1}B & \ldots & a_{mn}B \\ \end{bmatrix} \begin{bmatrix} x_{1}y \\ \vdots \\ x_{n}y \\ \end{bmatrix}\\ &= \begin{bmatrix} \sum_{j=1}^na_{1j}x_{j}By \\ \vdots \\ \sum_{j=1}^na_{mj}x_{j}By \\ \end{bmatrix} \\ &= \begin{bmatrix} \sum_{j=1}^na_{1j}x_{j} \\ \vdots \\ \sum_{j=1}^na_{mj}x_{j} \\ \end{bmatrix} \otimes By \\ &= Ax \otimes By \end{align}

Answer 3

This is my answer for (A$\otimes$B)$^\dagger$ = A$^\dagger$$\otimes$B$^\dagger$:

\begin{align} (A \otimes B)^\dagger &= \begin{bmatrix} a_{11}B &\ldots&a_{1n}B \\ \vdots & \ddots & \vdots\\ a_{m1}B & \ldots & a_{mn}B \\ \end{bmatrix} ^\dagger \\ &= \begin{bmatrix} \overline{a_{11}}B^\dagger &\ldots&\overline{a_{m1}}B^\dagger \\ \vdots & \ddots & \vdots\\ \overline{a_{1n}}B^\dagger & \ldots & \overline{a_{mn}}B^\dagger \\ \end{bmatrix} \\ &= \begin{bmatrix} \overline{a_{11}} &\ldots&\overline{a_{m1}} \\ \vdots & \ddots & \vdots\\ \overline{a_{1n}} & \ldots & \overline{a_{mn}} \\ \end{bmatrix} \otimes B^\dagger \\ &= A^\dagger \otimes B^\dagger \end{align}

I found this paper by Bobbi Jo Broxson, which confirms my results, and has several more properties on Tensor product properties included.