I want to "design" (build) a matrix that have multiple known eigenvalues. For these apriori chosen eigenvalues, I want to know the corresponding eigenvectors, too.
The point is that I want to start from the eigenvalues, chose the eigenvectors appropriately and then compute the matrix. All values must be integers.
I need this for a cryptographic application. Namely, I'm looking for something similar to Chinese remainder theorem, but done with eigenvalues and eigenvectors. To make an idea, in paper Fully Homomorphic SIMD Operations the authors propose an improvement upon previous schemes. They choose a specific cyclotomic polynomial which factorizes in n polynomials and so they can encrypt n message at a time. I want the same but for the approximate eigenvector problem.
You say you want to freely choose eigenvalues and eigenvectors. So you've chosen eigenvalues $\lambda_1,\ldots,\lambda_n$ and respective eigenvectors $\vec v_1,\ldots,\vec v_n$. You want a matrix $A\in\mathbb R^{n\times n}$ for which $A\vec v_i = \lambda_i\vec v_i$ for $i=1,\ldots,n$.
Let $V= \big[ \vec v_1,\ \ldots\ ,\ \vec v_n \big] \in \mathbb R^{n\times n}$. Let $$ \Lambda = \operatorname{diag}(\lambda_1,\ldots,\lambda_n) = \begin{bmatrix} \lambda_1 \\ & \lambda_2 \\ & & \lambda_3 \\ & & & \ddots \\ & & & & \lambda_n \end{bmatrix}. $$
Then $$ A = V \Lambda V^{-1} $$ will do the trick. Observe that $$ V^{-1} \vec v_i = \begin{bmatrix} 0 \\ \vdots \\ 0 \\ 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix} = \vec e_i \qquad \text{(with a 1 only in the $i$th position)}, $$ and $$ \Lambda \vec e_i = \begin{bmatrix} 0 \\ \vdots \\ 0 \\ \lambda_i \\ 0 \\ \vdots \\ 0 \end{bmatrix} = \lambda_i \vec e_i \qquad \text{(with $\lambda_i$ only in the $i$th position)}, $$ and finally $$ V(\lambda_i \vec e_i) = \lambda_i \vec v_i, $$ so $$ (V\Lambda V^{-1}) \vec v_i = \lambda_i \vec v_i. $$