Definition: A square matrix is diagonalizable if there exist matrices such that is invertible, is diagonal (has only entries on the main diagonal), and .
Problem: Solve the differential equation where is a diagonalizable square matrix and is a vector valued function (which must have components).
Solution: Well this is a lot like the regular differential equation which has solution . So let’s guess that
is the solution. But what exactly does mean? Consider the following.
Definition: If is a function and is a diagonal square matrix, then we say that is simply the matrix with applied to all the elements on the diagonal.
If is diagonalizable, then we have and we define .
So does work as a solution to the differential equation? It turns out that, by using the definition of the derivative we can show that
However, if we use the function , we get
as desired. QED.
Comment: Applying functions to matrices is a super cool thing and being able to solve differential equations with matrix coefficients is even better. Linear algebra provides all sorts of new ways to work with matrices.
Practice Problem: Show that all symmetric square matrices are diagonalizable.
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