Fortunately, we need not rely on the procedure of Theorem EMHE each time we need an eigenvalue. It is the determinant, and specifically Theorem SMZD, that provides the main tool for computing eigenvalues. Here is an informal sequence of equivalences that is the key to determining the eigenvalues and eigenvectors of a matrix, \begin{equation*} A\vect{x}=\lambda\vect{x}\iff A\vect{x}-\lambda I_n\vect{x}=\zerovector\iff \left(A-\lambda I_n\right)\vect{x}=\zerovector \end{equation*}
So, for an eigenvalue $\lambda$ and associated eigenvector $\vect{x}\neq\zerovector$, the vector $\vect{x}$ will be a nonzero element of the null space of $A-\lambda I_n$, while the matrix $A-\lambda I_n$ will be singular and therefore have zero determinant. These ideas are made precise in Theorem EMRCP and Theorem EMNS, but for now this brief discussion should suffice as motivation for the following definition and example.
Suppose that $A$ is a square matrix of size $n$. Then the characteristic polynomial of $A$ is the polynomial $\charpoly{A}{x}$ defined by \begin{equation*} \charpoly{A}{x}=\detname{A-xI_n} \end{equation*}
The characteristic polynomial is our main computational tool for finding eigenvalues, and will sometimes be used to aid us in determining the properties of eigenvalues.
Suppose $A$ is a square matrix. Then $\lambda$ is an eigenvalue of $A$ if and only if $\charpoly{A}{\lambda}=0$.
Let us now turn our attention to the computation of eigenvectors.
Suppose that $A$ is a square matrix and $\lambda$ is an eigenvalue of $A$. Then the eigenspace of $A$ for $\lambda$, $\eigenspace{A}{\lambda}$, is the set of all the eigenvectors of $A$ for $\lambda$, together with the inclusion of the zero vector.
Example SEE hinted that the set of eigenvectors for a single eigenvalue might have some closure properties, and with the addition of the one eigenvector that is never an eigenvector, $\zerovector$, we indeed get a whole subspace.
Suppose $A$ is a square matrix of size $n$ and $\lambda$ is an eigenvalue of $A$. Then the eigenspace $\eigenspace{A}{\lambda}$ is a subspace of the vector space $\complex{n}$.
Theorem EMS tells us that an eigenspace is a subspace (and hence a vector space in its own right). Our next theorem tells us how to quickly construct this subspace.
Suppose $A$ is a square matrix of size $n$ and $\lambda$ is an eigenvalue of $A$. Then \begin{equation*} \eigenspace{A}{\lambda}=\nsp{A-\lambda I_n} \end{equation*}
You might notice the close parallels (and differences) between the proofs of Theorem EMRCP and Theorem EMNS. Since Theorem EMNS describes the set of all the eigenvectors of $A$ as a null space we can use techniques such as Theorem BNS to provide concise descriptions of eigenspaces. Theorem EMNS also provides a trivial proof for Theorem EMS.