# Linear algebra - eigenvalues, eigenvectors

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

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