Most high-school science programs have a strong focus on scientific theory and do not train students to conduct independent research. Previous work has demonstrated the efficacy of a mentor-supported, student-driven teaching program to effectively introduce research-specific skills in a classroom context. Despite the effectiveness of such programs, their class-based formats and requirements for multiple full-time faculty mentors limit their throughput, and the finite expertise of full-time mentors requires participants to focus on specific research subjects.

Let’s compute what \(e^{tA}\) should look like, where \(t\) is some scalar and \(A\) is a diagonalizable matrix. This is a supplement to Second-Order Linear Differential Equations.

Let \(v_1\dots v_{m}\) be the eigenvectors of \(A\). Let \(\lambda_{1}\dots\lambda_{m}\) be the eigenvalues.

Recall that we can therefore diagonalize \(A\) as:

\begin{equation} A = \mqty(v_1& \dots& v_{m})\mqty(\dmat{\lambda_{1}, \dots, \lambda_{m}})\mqty(v_1& \dots& v_{m})^{-1} \end{equation}

read: change of choordinates into the eigenbases, scale by the eigenvalues, then change back to normal choordinates.