Why is this "not allowed by definition"? Eigenvectors and Eigenvalues I have just started learning about eigenvectors and eigenvalues, and encountered the following (unjustified) statement. $ (A - \lambda I)\mathbf{v} = 0$. If the matrix $ (A - \lambda I) $ were invertible, then the solution would simply be $\mathbf{ v } = 0$. But this is not allowed by definition. I'm curious as to why it is "not allowed by definition"? I would appreciate it if someone could elaborate on the reasoning behind this. Thank you.

The eigenvalue equation can also be written as $Av=\lambda v $, so we are looking for vectors $v$, on which the operator $A$ acts by scalar multiplication. These vectors usually say some interesting things about the nature of the operator $A$, as well as helps with solving problems. After all, many linear operator-related problems are far easier if the operator is diagonal, and the diagonalization of operators are done via finding the eigenvalues/eigenvectors.

The thing is, if $v=0$, then the equation $Av=\lambda v$ is **always** satisfied, regardless of what $A$ is. Therefore, although this is a solution of the equation, it is not an interesting solution, since the whole point of solving an eigenvalue equation is to understand the structure of the operator $A$. If something is true for all $A$s, then it must contain no information about specific $A$s.