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Martin Sleziak
Martin Sleziak
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Closed form of the product of these $2\times 2$ matriesmatrices

I have a series of $2\times 2$ matriesmatrices denoted by $$ M_j=\begin{pmatrix}e^{i\theta} & 0 \\ 0 & e^{-i\theta}\end{pmatrix}+a_j\begin{pmatrix}-e^{i\theta} & e^{i\theta} \\ e^{-i\theta} & e^{-i\theta}\end{pmatrix} $$ where $\theta$ is real, and $a_j$ is non-zero complex numbers. I want to find out $$ S=\prod_j^NM_j. $$ Is it possible to obtain the explicit expression of $S$?

Closed form of the product of these $2\times 2$ matries

I have a series of $2\times 2$ matries denoted by $$ M_j=\begin{pmatrix}e^{i\theta} & 0 \\ 0 & e^{-i\theta}\end{pmatrix}+a_j\begin{pmatrix}-e^{i\theta} & e^{i\theta} \\ e^{-i\theta} & e^{-i\theta}\end{pmatrix} $$ where $\theta$ is real, and $a_j$ is non-zero complex numbers. I want to find out $$ S=\prod_j^NM_j. $$ Is it possible to obtain the explicit expression of $S$?

Closed form of the product of these $2\times 2$ matrices

I have a series of $2\times 2$ matrices denoted by $$ M_j=\begin{pmatrix}e^{i\theta} & 0 \\ 0 & e^{-i\theta}\end{pmatrix}+a_j\begin{pmatrix}-e^{i\theta} & e^{i\theta} \\ e^{-i\theta} & e^{-i\theta}\end{pmatrix} $$ where $\theta$ is real, and $a_j$ is non-zero complex numbers. I want to find out $$ S=\prod_j^NM_j. $$ Is it possible to obtain the explicit expression of $S$?

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Guoqing
Guoqing
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Closed form of the product of these $2\times 2$ matries

I have a series of $2\times 2$ matries denoted by $$ M_j=\begin{pmatrix}e^{i\theta} & 0 \\ 0 & e^{-i\theta}\end{pmatrix}+a_j\begin{pmatrix}-e^{i\theta} & e^{i\theta} \\ e^{-i\theta} & e^{-i\theta}\end{pmatrix} $$ where $\theta$ is real, and $a_j$ is non-zero complex numbers. I want to find out $$ S=\prod_j^NM_j. $$ Is it possible to obtain the explicit expression of $S$?