  Diagonalizing a matrix $A$, that is finding two matrices $P$ and $D$ such
that $A = PDP^{-1}$ with $D$ being a diagonal matrix needs two steps: first
find the eigenvalues and then find the corresponding eigenvectors. We show that
we do not need the second step when diagonalizing matrices with a spectrum,
$\left|\sigma(A)\right|\leq 2$ since those vectors already appear as nonzero
columns of the $\textit{eigenmatrices}$, a term defined in this work. We
further generalize this for matrices with $\left|\sigma(A)\right|> 2$ and show
that eigenvectors lie in the column spaces of eigenmatrices of the
complementary eigenvalues, an approach without using the classical Gauss-Jordan
elimination of rows of a matrix.
  We introduce two major results, namely, the $\textit{2-Spectrum Lemma}$ and
the $\textit{Eigenmatrix Theorem}$. As a conjecture, we further generalize the
Jordan canonical forms for a new class of generalized eigenvectors that are
produced by repeated multiples of certain eigenmatrices. We also provide
several shortcut formulas to find eigenvectors that does not use echelon forms.
The method discussed in this work may be summarized with the mnemonic "Find
your puppy at your neighbors'!" argument, where puppy is the eigenvector and
the neighbors are the complementary eigenmatrices.
