Linear Algebra with Differential Equations/Homogeneous Linear Differential Equations/Repeated Eigenvalue Method

When the eigenvalue is repeated we have a similar problem as in normal differential equations when a root is repeated, we get the same solution repeated, which isn't linearly independent, and which suggest there is a different solution. Because the case is very similar to normal differential equations, let us try ${\displaystyle \mathbf {X} =\mathbf {u} te^{\mathbf {\lambda } t}}$ for ${\displaystyle \mathbf {X} '=\mathbf {A} \mathbf {X} }$ and we see that this does not work; however, ${\displaystyle \mathbf {X} =(\mathbf {B} t+\mathbf {C} )e^{\mathbf {\lambda } t}}$ DOES work (For the observant reader, this gives a hint to the changes in the Method of Undetermined Coefficients as compared to differential equations without linear algebra).
In fact if we use this we see that ${\displaystyle \mathbf {B} =\mathbf {u} }$ where ${\displaystyle \mathbf {u} }$ is a typical eigenvector; and we see that ${\displaystyle \mathbf {C} =\mathbf {n} }$ where ${\displaystyle \mathbf {n} }$ is a normal eigenvector defined by ${\displaystyle (\mathbf {A} -\lambda \mathbf {I} )\mathbf {C} =\mathbf {B} }$
Thus our fundamental set of solutions is: ${\displaystyle \{\mathbf {u} te^{\lambda t}+\mathbf {n} e^{\lambda t};\mathbf {u} e^{\lambda t}\}}$