Let $n \times n $ matrix A has pairwise distinct eigenvalues $\alpha_{1}, .. \alpha_{n}$ and eigenvectors $a_{1}, ..., a_{n}$, also matrix B $m\times m$ has pairwise distinct eigenvalues $\beta_{1}, .. \beta_{m}$ and eigenvectors $b_{1}, ..., b_{m}$. Find eigenvalues and eigenvectors for operators: $$S: Mat_{n\times m} \to Mat_{n\times m}, S(X) = AX + XB^T; $$ $$P: Mat_{n\times m} \to Mat_{n\times m}, P(X) = AXB^T;$$ please help, I would be very grateful if you could explain to me the approach to such problems and suggest the literature for such problems

  • $\begingroup$ Why are you interested in this problem and what have you tried? $\endgroup$ – James Mar 6 at 10:52
  • $\begingroup$ You will find an approach in the answer to this question: math.stackexchange.com/questions/3118853/… $\endgroup$ – James Mar 6 at 10:54
  • $\begingroup$ James, I supposed that $X = a_{i} \times b_{i}, where $ $a_{i} \in R^n and$ $ b_{i} \in R^m$, then $S(a_{i}\times b_{I}^T) = A(a_{i}\times b_{i}^T) + (a_{i}\times b_{i}^T)B^T= \alpha_{i}a_{i}b_{I}^T+\beta_{i}a_{i}b_{i}^T = ( \alpha_{I}+ \beta_{i})a_{i}b_{I}^T$ $\endgroup$ – GIFT Mar 6 at 11:43
  • $\begingroup$ I am interested in this problem, because I want to understand the main idea of eigenvectors and eigenvalues. $\endgroup$ – GIFT Mar 6 at 11:51

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