Exercise 7.3

We are going to prove, step by step, the following equation:

\[ (A \otimes B) \, ( a \otimes b) = (Aa \otimes Bb) \]

a)

We replace the symbols with their component form:

\[ \left( \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} \otimes \begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{pmatrix} \right) \left( \begin{pmatrix} a_{11} \\ a_{21} \end{pmatrix} \otimes \begin{pmatrix} b_{11} \\ b_{12} \end{pmatrix} \right) = \left( \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} \begin{pmatrix} a_{11} \\ a_{21} \end{pmatrix} \otimes \begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{pmatrix} \begin{pmatrix} b_{11} \\ b_{12} \end{pmatrix} \right) \]

b)

We perform the matrix multiplication on the right-hand side:

\[ \left( \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} \otimes \begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{pmatrix} \right) \left( \begin{pmatrix} a_{11} \\ a_{21} \end{pmatrix} \otimes \begin{pmatrix} b_{11} \\ b_{12} \end{pmatrix} \right) = \left( \begin{pmatrix} A_{11} \, a_{11} + A_{12} \, a_{21} \\ A_{21} \, a_{11} + A_{22} \, a_{21} \end{pmatrix} \otimes \begin{pmatrix} B_{11} \, b_{11} + B_{12} \, b_{21} \\ B_{21} \, b_{11} + B_{22} \, b_{21} \end{pmatrix} \right) \]

c)

We expand all three Kronecker products: \[ \left( \begin{pmatrix} A_{11}B_{11} & A_{11}B_{12} & A_{12}B_{11} & A_{12}B_{12} \\ A_{11}B_{21} & A_{11}B_{22} & A_{12}B_{21} & A_{12}B_{22} \\ A_{21}B_{11} & A_{21}B_{12} & A_{22}B_{11} & A_{22}B_{12} \\ A_{21}B_{21} & A_{21}B_{22} & A_{22}B_{21} & A_{22}B_{22} \end{pmatrix} \begin{pmatrix} a_{11} b_{11} \\ a_{11} b_{21} \\ a_{21} b_{11} \\ a_{21} b_{21} \\ \end{pmatrix} \right) = \begin{pmatrix} (A_{11} \, a_{11} + A_{12} \, a_{21})(B_{11} \, b_{11} + B_{12} \, b_{21} ) \\ (A_{11} \, a_{11} + A_{12} \, a_{21}) (B_{21} \, b_{11} + B_{22} \, b_{21}) \\ (A_{21} \, a_{11} + A_{22} \, a_{21}) (B_{11} \, b_{11} + B_{12} \, b_{21} ) \\ (A_{21} \, a_{11} + A_{22} \, a_{21}) (B_{21} \, b_{11} + B_{22} \, b_{21}) \end{pmatrix} \]

d)

\(A \otimes B\): 4 x 4 matrix

\(a \otimes b\): 4 x 1 matrix

\(Aa \otimes Bb\): 4 x 1 matrix

e)

We perform the matrix multiplication on the left-hand side:

\[ \begin{pmatrix} A_{11}B_{11} \, a_{11} b_{11} + A_{11}B_{12} \, a_{11} b_{21} + A_{12}B_{11} \, a_{21} b_{11} + A_{12}B_{12} \, a_{21} b_{21} \\ A_{11}B_{21} \, a_{11} b_{11} + A_{11}B_{22} \, a_{11} b_{21} + A_{12}B_{21} \, a_{21} b_{11} + A_{12}B_{22} \, a_{21} b_{21} \\ A_{21}B_{11} \, a_{11} b_{11} + A_{21}B_{12} \, a_{11} b_{21} + A_{22}B_{11} \, a_{21} b_{11} + A_{22}B_{12} \, a_{21} b_{21} \\ A_{21}B_{12} \, a_{11} b_{11} + A_{21}B_{22} \, a_{11} b_{21} + A_{22}B_{21} \, a_{21} b_{11} + A_{22}B_{22} \, a_{21} b_{21} \end{pmatrix} = \begin{pmatrix} (A_{11} \, a_{11} + A_{12} \, a_{21})(B_{11} \, b_{11} + B_{12} \, b_{21} ) \\ (A_{11} \, a_{11} + A_{12} \, a_{21}) (B_{21} \, b_{11} + B_{22} \, b_{21}) \\ (A_{21} \, a_{11} + A_{22} \, a_{21}) (B_{11} \, b_{11} + B_{12} \, b_{21} ) \\ (A_{21} \, a_{11} + A_{22} \, a_{21}) (B_{21} \, b_{11} + B_{22} \, b_{21}) \end{pmatrix} \]

f)

Finally we multiply the brackets on the right hand side and get a sum of four terms in each row:

\[ \begin{pmatrix} A_{11}B_{11} \, a_{11} b_{11} + A_{11}B_{12} \, a_{11} b_{21} + A_{12}B_{11} \, a_{21} b_{11} + A_{12}B_{12} \, a_{21} b_{21} \\ A_{11}B_{21} \, a_{11} b_{11} + A_{11}B_{22} \, a_{11} b_{21} + A_{12}B_{21} \, a_{21} b_{11} + A_{12}B_{22} \, a_{21} b_{21} \\ A_{21}B_{11} \, a_{11} b_{11} + A_{21}B_{12} \, a_{11} b_{21} + A_{22}B_{11} \, a_{21} b_{11} + A_{22}B_{12} \, a_{21} b_{21} \\ A_{21}B_{12} \, a_{11} b_{11} + A_{21}B_{22} \, a_{11} b_{21} + A_{22}B_{21} \, a_{21} b_{11} + A_{22}B_{22} \, a_{21} b_{21} \end{pmatrix} = \begin{pmatrix} A_{11} \, a_{11} B_{11} \, b_{11} + A_{11} \, a_{11} B_{12} \, b_{21} + A_{12} \, a_{21} B_{11} \, b_{11} + A_{12} \, a_{21} B_{12} \, b_{21} \\ A_{11} \, a_{11} B_{21} \, b_{11} + A_{11} \, a_{11} B_{22} \, b_{21} + A_{12} \, a_{21} B_{21} \, b_{11} + A_{12} \, a_{21} B_{22} b_{21} \\ A_{21} \, a_{11} B_{11} \, b_{11} + A_{21} \, a_{11} B_{12} \, b_{21} + A_{22} \, a_{21} B_{11} \, b_{11} + A_{22} \, a_{21} B_{12} \, b_{21} \\ A_{21} \, a_{11} B_{12} \, b_{11} + A_{21} \, a_{11} B_{22} \, b_{21} + A_{22} \, a_{21} B_{21} \, b_{11} + A_{22} \, a_{21} B_{22} \, b_{21} \end{pmatrix} \]