How to construct a matrix with given vectors and scalars as eigenvectors and eigenvalues?

Question

I don't have an example but just wondering what if we are given a list of scalars and vectors, then how to write a matrix corresponding to those vectors and scalars as eigenvectors and eigenvalues using Jordan Canonical Form? Can someone give me some ideas on how to do this ? I want to know the specific steps applying Jordan Canonical Form.

Answer 1

Essentially, you want to extend you vectors to a basis, write J.C.F. for the standard basis, then convert between the basis, and use this conversion to convert from J.C.F. to the matrix you want.

Suppose $v_1,\cdots,v_k$ are linearly independent column vectors in $\Bbb{R}^n$ and $\lambda_1,\cdots,\lambda_k$ are scalars. Extend $v_1,\cdots,v_k$ to a basis $v_1,\cdots,v_n$ for $\Bbb{R}^n$. Say $\{e_i\}$ is the standard basis for $\Bbb{R}^n$. Form a matrix $A$ for a linear transformation which sends each $v_j$ to a standard basis vector $e_{i_j}$ (To make it easier, say that $i_{j_1} < i_{j_2}$ whenever $1 \leq j_1 <j_2 \leq k$). Let $P = \begin{bmatrix} J_{\lambda_1,i_1}&&&\\ & J_{\lambda_2,i_2-i_1}&&\\ &&\ddots&\\ &&&J_{\lambda_k,i_k-i_{k-1}} \end{bmatrix}$. Then, your matrix is $A^{-1}PA$.